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" But that which relies on calculation and 
measurement will be the best element in the soul?" 

" Of course." 

11 Then that which opposes it will be one of the 
beggarly elements in us? 

" Inevitably." 

PLATO, Republic. 

Christian Huygens 





A. E. BELL. Ph.D., M.Sc. 

Head of the Science Department, Sandhurst 
Formerly Head of the Science Department, Clifton College, 




First published 1947 
Reprinted 1950 

Printed in Great Britain by 

Sons Ltd., Guild ford and Es/ier 


THERE can be no doubt that Christian Huygens was one of the 
greatest scientific geniuses of all time. A man who transformed 
the telescope from being a toy into a powerful instrument of in- 
vestigation, and this as a consequence of profound optical 
researches; who discovered Saturn's ring and the satellite Titan; 
who drew attention to the Nebula in Orion; who studied the prob- 
lem of gravity in a quantitative manner, arriving at correct ideas 
about the effects of centrifugal force and the shape of the earth; 
who, in the great work Horologium Oscillatorium, founded the 
dynamics of systems and cleared up the whole subject of the 
compound pendulum and the tautochrone; who solved the out- 
standing problems concerned with collision of elastic bodies and 
out of much intractable work developed the general notion of 
energy and work; who is rightly regarded as the founder of the 
wave theory in light, and thus of physical optics such a man 
deserves memory with the names of Galileo and Newton, and 
only the accidents of history have so far prevented this. It might 
be argued that Robert Hooke, who like Huygens was influenced 
by Descartes 's teachings, is of equal stature and showed as much 
inventive genius and intuition. In the extraordinary range of 
their activities there is some similarity. The overwhelming 
difference lies in the fact that Huygens was a great mathema- 
tician and exponent of the quantitative method, whereas Hooke 
could never get beyond the first phase of a piece of work : that 
which led to the need of exact measurement and the discovery of 
mathematical relations. 

Having made this claim for Huygens, it is natural to ask how 
he compares with Newton. It is a question which arises from 
time to time in these pages, and one to which there is no 
epigrammatic answer. In some senses it was Huygens's greatest 
misfortune to grow up under the powerful influence of Descartes, 
who was a grfeat friend of his father, a frequent visitor to the 



family, and master of at least one disciple who taught Huygens 
at the university. From Descartes too many of Huygens's 
hypotheses were taken, so much so that he might stand as the 
exact opposite of Newton, whose objection hypotheses non fingo 
he did so much to call forth. Looking at Huygens in this way it 
is easy to dismiss him as a " Cartesian ", one whose ideas were 
largely superseded after the publication of Newton's Principia. 
But this would be a serious mistake. If he could not so 
brilliantly see the way to extend the sphere of natural law to 
the heavens, perceiving that the task of science is not to 
disclose a mechanism so much as to arrive at laws, he neverthe- 
less did important work to prepare men of science for this 
modern attitude. If Newton owed nothing to Huygens, and he 
certainly owed exceedingly little, it is very probable that he was 
indebted in another way, for it may well have been the feeling of 
dissatisfaction with the position men like Huygens were reaching 
that drove Newton to make the new " instauration " Bacon had 
looked for a renovation of natural philosophy. The progress 
of scientific explanation may then be seen to be a process of 
leaving out redundant elements, of emancipation from imaginary 
qualities, until one arrives at the really successful procedure of 

But Huygens was in all other senses an astoriishingly modern 
thinker, and he had the disposition which sets out to face things 
as they are which marks the man of science as much as does the 
possession of specialized knowledge. As a scientific researcher 
he was the first of a new profession, and this permanent occupa- 
tion with science brought its own characteristic attitude of mind. 
Colbert, Louis XIV's energetic and shrewd minister, subsidized 
scientific investigation for the first time in history when he 
obtained pensions for Huygens and the other scientists who 
formed the nucleus of the Academic Royale des Sciences. Of 
course Colbert had his eyes on commercial as much as on 
intellectual advances. Considering his jealousy of Dutch com- 
merce Colbert was indeed fortunate to get as an ally of French 
power the most noted Dutch scientist of the age. It has indeed 
been a puzzling feature of Huygens's life that, having embraced 
French manners, delighting in the freedom that his position at 
the Biblioth&jue du Roi gave him, he could even so remain in 
Paris in his official position during the years when Louis waged 
war on the Netherlands, determined if he could to break the 


newly found independence of the Dutch for ever. More than 
this, Huygens in 1673 dedicated his magnum opus, the Horolo- 
gium Oscillatorium, to his royal patron. 

Huygens possessed a delicate constitution and was, it should 
be admitted, not of the stuff of which fighters are made. Like 
Pascal he suffered from frequent illness, like Spinoza there was a 
certain effeminacy about him. Again it may be argued that war 
in those days never concerned more than the limited class of 
professional soldier. If Huygens had quitted Paris the best he 
could have done towards the defence of Holland would have 
been to work in a diplomatic capacity as did his brother 
Constantin, or decipher codes as did the mathematician John 
Wallis in the civil war in England, or be killed like any ordinary 
soldier as was Gascoigne at Marston Moor. Men of the profes- 
sional class in those days were not expected to give up their 
activities, and there can be little doubt that Huygens's scientific 
work benefited greatly from his life in Paris. At home at 
Voorburg, near the Hague, he lived the life of a recluse with only 
this stimulus of his correspondence with Paris and London. The 
investigation of the physical world appeared to Huygens and to 
nearly all men of science to be something of such moment that all 
events in the social and political world were merely so many 
inconvenient interruptions. That this was so in the seventeenth 
century is evident from the way in which the meetings which led 
to the foundation of the Royal Society were carried on during the 
civil war. From the very beginning the men of science felt them- 
selves to be advancing the real causes of humanity and this 
longer view may well have been the one at which Huygens looked 
when, in 1672, he decided to remain in Paris. 

From birth Christian Huygens grew up under the strongest 
French influences. In Paris, as a young man, he evidently 
imbibed the fashionable ideas which in religion tended to 
agnosticism and in morals pointed the way to greater freedom. 
His seriousness forbade the lax immorality then current. " The 
beaux esprlts believed in God merely as a matter of courtesy and 
for reasons of state ", wrote Garasse, but while this might be 
true, the men of science, especially in England, kept their 
religion. Huygens stands in contrast with the other great scien- 
tists of his time and in closer resemblance with some of the 
encyclopaedists of the following century in that he turned away 
from Calvinism as from Catholicism. 


In spite of the evident growth of power politics and the 
existence of widespread corruption, the period of Huygens's life 
was one of extraordinary optimism among the men of science. 
Science was the last activity to come from the humanistic 
impulse and it was to its devotees the most dazzling discovery to 
be attributed to man's freedom. "... this is the age ", wrote 
Henry Power, " wherein all men's souls are in a kind of fermen- 
tation, and the spirit of wisdom and learning begins to mount 
and free itself from those drossie and terrene Impediments 
where-with it hath been so long clogg'd . . . This is the age 
wherein (methinks) philosophy comes in with a Spring-tide ..." 
The greater minds of the period were less confident over the in- 
evitability of progress; Huygens in particular was especially 
cautious when asked to compare his own times with the age of 
Pericles. But a cruder spirit came to be associated with the men 
of science of the eighteenth and nineteenth centuries. 

At least one modern writer has, however, condemned the 
" false modern emphasis " on the bold confidence and rebellious 
energy of the Renaissance, and has pointed out the amount of 
pessimism in English seventeenth century literature. On the 
Continent the inroads on religious belief seem in fact to have 
been more rapid than they were in England. Bishop Sprat, 
writing a defence of the Royal Society in 1667, believed that 
science would never undermine the socially acceptable beliefs of 
his time. He never dreamed that a "Universal Inquiry into things 
hitherto unquestioned " could have the unending consequences 
it has in fact produced. He obviously supposed, as did Descartes, 
though not Huygens, that the complete scientific account of the 
universe would in its essentials require the labours of only one or 
two generations of gifted intelligences. In these labours the 
experimentalists would, thought Sprat, have always before their 
eyes the " Beauty, Contrivance and Order of God's works ". As 
the Master of Trinity College has written, " God was to be 
praised by studying the plan of His creation, but no further 
attempt was to be made to fit the findings of science into the 
scheme of theology, as the schoolmen of old had striven so long 
and painfully to do." This was Newton's position, it was 
Boyle's, but it was not Huygens's. He alone among the men of 
science of his day found the temper of scientific enquiry alien to 
orthodox religious faith. As it was, both in France and England 
I here were divines who supported the plan of organized enquiry 


in natural philosophy. Liberal minded abbes and protestant 
bishops gave their approval to scientific research for the glory of 
God and the service of man. Occasionally the former tended to 
lose their nerve and clung then for stability to Descartes's 
" system ". 

The particular feature of Huygens's thought which was at 
the same time his strength and weakness was his concern for 
particular problems and his distrust of all speculative general- 
izations. This distrust he no doubt acquired during his study of 
Descartes's writings. His attitude towards Spinoza's ideas may be 
explained in this way, for it would otherwise have been expected 
that these ideas would have gained his sympathy, for Spinoza, 
of all great religious writers, has come nearest to expressing the 
scientific attitude to the world. But Huygens distrusted the 
system which Spinoza elaborated and no doubt for the reason 
that Spinoza sought to apply the Cartesian method. This 
method, Huygens saw, is unable to give us an understanding of 
nature, and he himself made great contributions to the new 
scientific method. The fact that he personally despised Spinoza 
seems to be explained by a sense of superiority which rested on 
social rather than on intellectual considerations. It was not, of 
course, a democratic age. 

In his scientific work Huygens was the greatest mechanist of 
the seventeenth century. He combined Galileo's mathematical 
treatment of phenomena with Descartes's vision of the ultimate 
design of nature. Beginning as an ardent Cartesian who sought 
to correct the more glaring errors of the system, he ended as one 
of its sharpest critics. The development in the seventeenth 
century of Dynamics and Astronomy beyond the stage of 
geometrical description demanded new inductive principles of 
correlation; the ideas of mass, weight, momentum, force, and 
work were finally clarified in Huygens's treatment of the phen- 
omena of impact, centripetal force and the first dynamical 
system ever studied the compound pendulum. In Astronomy 
Huygens explained the appearances of Saturn, until 1656 the 
greatest anomaly of the Copernican system. His eminence as an 
observer was due to the high quality of his telescopes and this, in 
part, resulted from his thorough theoretical researches on the 
problems first attacked by Kepler, Snell and Descartes. It is well 
known that physical optics practically took its rise from 
Huvgens's Traite de la Lumi&re. 


The growth of interest in the history of science may be con- 
sidered to be in a sense an outcome of the increasing specializa- 
tion of science itself. So much is commonly ignored in concen- 
trating on the discipline of science, that for education in the 
broader sense, when once the demands of life are allowed to 
supersede merely professional ones, something more is needed. 
Many have felt that the history of science may provide an im- 
portant humanistic element. A protest may here be made against 
the practice of inserting historical notes in scientific textbooks 
without regard for the conflict of old and modern ideas. Once a 
subject has become well developed, the logical and not the histor- 
ical method is to be desired, for so much of the earlier science can 
be properly understood only with a really adequate study. The 
great scientist of the past tends soon to appear a distant and 
indeed thoroughly dead sort of figure. The modern scientist to 
a large extent accepts his reputation on trust and has little time 
and often less inclination to read the original work. How many 
now read Galileo's Discourses or Newton's Principiaf It is other- 
wise in "art and in literature. If science is to become a more 
widely accepted means of education (in the sense of a form of 
culture) there is need of more works on its history. A modern 
estimate can do much to revivify the past and present these great 
men, its creators, in a clearer light. 

As an account in English of the work of Huygens this study 
is to be regarded as only a beginning. The Oeuvres Completes de 
Christiaan Huygens, published by the Societe Hollandaise des 
Sciences, comprise more than twenty magnificent volumes and 
contain all the material for a definitive work; beside them the 
present book must appear almost insignificant. It is greatly to be 
hoped that before long a large work on this great subject will be 
written by a scholar of the requisite stature. Writing as one whose 
training has been principally in chemistry the author has met with 
many difficulties in Huygens's works. It need hardly be said that 
they are properly to be studied by a mathematician, while the 
subject as a whole requires a combination of historian and Latin 
scholar, physicist and philosopher, which it must be rare to 


CLIFTON, 1947 













The Horologium Oscillatorium 
V. Part I. Construction and Use of the 

Pendulum Clock . . . . 127 

VI. Part II. Oscillation in a Cycloidal Arc . . 136 

VII. Part III. Evolutes and the Measurement of 

Curves 145 

VIII. Part IV. The Centre of Oscillation of a Com- 
pound Pendulum . . . . 150 









INDEX 219 



Christian Huygens frontispiece 

facing page 
I. Saturn reproduced from Huygens's MS 32 

II. One of Coster's clocks 38 

III. Huygens's Clock as the centre feature of a design 

showing scientific apparatus of 1671 56 

IV. Members of the Academic Royalc dcs Sciences 58 
V. Louis XIV at a meeting of the Academic 60 

VI. Drawing by Huygens of his vacuum pump, 1668 162 


CHRISTIAN HUYGENS has been a strangely neglected figure 
apart from the study he has rightly received in his native 
Holland. A man of the greatest scientific genius without any 
doubt, he was one in whom great sagacity and mathematical 
power went side by side with a feeling for elegance and form in 
the interpretation we make of Nature, so much so, that it is with- 
out surprise that we find he was devoted to music and the arts 
ajid was by no means the type of narrow research worker that 
later scientific studies did for a time produce, and still produce 
in some measure. Huygens was a professional scientist in an age 
when the boundaries of Science were scarcely drawn, and his 
interest lies as much in his general outlook as in his specialized 

Huygens had not the religious feeling of a Spinoza or the 
sensitivity of a Pascal, nor was he a philosopher of the stature of 
Descartes or a mathematician of the rank of Leibnitz. In an age 
when the human mind was making great marches into the 
territory of natural philosophy, Huygens's energies were thrown 
now into the study of applied mathematics, now into optical 
researches or astronomy; and he managed somehow to pursue 
the most strikingly original researches in several subjects quite 
simultaneously, so that in his note-books matters of the most 
varied kind jostle one another in profusion, and a very large 
volume indeed would be needed to do justice to his labours. 
What is of chief significance to-day can be reduced to much 
smaller limits, and the reader who wishes for more must go to 
the great volumes published by the Societe Hollandaise des 
Sciences under the auspices of the Dutch Government. 

Here we are concerned rather to look back for a space on that 
interesting period in Europe between the death of Galileo in 



1642 and the rise to fame of Newton, a period in which 
Huygens, in fact, stood unchallenged as the greatest man of 
science of the age. 

It has been remarked 1 that " In 1600 the educated English- 
man's mind and world were more than half medieval; by 1660 
they were more than half modern ". And this remark need not 
have been limited to Englishmen. On the Continent also, about 
the middle of the century, a certain profoundly important 
change was becoming visible. It was, perhaps, in the years 
following 1670 that the break-away from authoritative teachings 
of Descartes as of the schoolmen became the feature of the 
really important scientific theories. Galileo and Huygens both 
struggled to make use of teaching they received in their youth, 
and both failed; they were each forced in some degree to rely on 
their own powers. Indeed, underneath all successful scientific 
work there lies a great deal of experiment in failure. 

One must read Dante, or toil over Thomas Aquinas, to get 
a picture of the universe as it was conceived by educated men in 
the Middle Ages. The sheer verbalism of all argument about the 
world repels and astonishes the modern reader, but there was an 
undeniable attractiveness in the notion of a Cosmos : the " idea 
of a hierarchically-ordered finite world structure ", a world in 
which all was made for man and consequently one in which 
clear and simple reasons existed why things are as they 
are. What we see as an appeal to objectivity must then have 
seemed to some to be pure obstinacy and blindness, for what the 
men of science really abolished was not so much an over-rational 
world structure as the appeal to feeling in the making of explana- 
tions. The new studies offered at first no more satisfaction than 
that which could be found in the agreement of theory with 
measurement. Nevertheless, scientific explanations did not get a 
reputation for their " inhuman " quality until the eighteenth 
century, when many physical theorems were generalized in 
abstract mathematical form. 

Early in the century Descartes worked out an ingenious and 
even aesthetically satisfying system which welded natural science 
on to the structure of a philosophical theory about the nature of 
matter and of space, and some reference to this system must be 
made in later pages. The chief point about Descartes's teaching, 

Douglas Bush : English Literature in the Earlier Seventeenth Century, 


if it were accepted, was that experiment and observation could 
soon be dispensed with and the human mind could rest satisfied 
with the knowledge it could gain through a rationale worked out 
by philosophers. So seductive was his reasoning, and so per- 
suasive the arrangement of the arguments, that both in France 
and England there were soon many ardent Cartesians who were 
distinguished by the ease with which they accounted (in a 
general way) for natural phenomena. Since space was supposed 
to be full of a " subtle matter " and this moved around each 
planet in a kind of vortex, it was easy to imagine various 
effects as resulting from the properties of this medium. And 
Huygens was himself for many years a Cartesian. The essays 
produced by Descartes were a flirtation with the mathematical 
treatment of observations begun by Galileo, only they went far 
further and cast the human mind in great voyages of imagina- 
tion further, in fact, than it was yet ready to go. It is always an 
interesting question, therefore, how Huygens came to be a strong 
critic of Cartesianism, and on the other hand, why he rejected 
Newton's treatment of gravitation and even at the end of his life 
had not thrown overboard the whole Cartesian apparatus. Of all 
the events in Huygens's life when one would give much to know 
what happened, there is an occasion of which one gets only a faint 
glimpse: Huygens and Newton getting into a stage coach at 
seven o'clock on a July morning in 1689, to go from Cambridge 
to London, Huygens was then sixty and his zeal and lively 
curiosity were unabated; Newton was forty-seven, and every- 
where acclaimed as the author of the magnificent Principia 
though it had to be confessed that only a handful of men really 
knew what it was all about. Huygens had left Holland in poor 
health in order to see Newton and to visit old friends among the 
English men of science. But all that can safely be connected with 
this meeting is the fact that Newton subsequently produced a 
further study of the Cartesian vortices and, on the other hand, 
Huygens began to object to Leibnitz's use of them. As the coach 
rolled on its way to London, it may be that Huygens was turn- 
ing over in his mind the final objections to any further develop- 
ments of Descartes's ideas. His own work had led far in this 
direction and the end of it all seemed to be that Descartes's 
ventures in physics had been pure romance, " un beau roman de 
physique " as Leibnitz himself concluded. Or is such conjecture 
too dangerous? Huygens, with his only moderate English and 


his weakness for a picturable sort of explanation may have made 
little of a taciturn Newton; he recorded nothing of interest from 
the meeting. 


Unlike Newton, Christian Huygens came of a family which 
had already shown genius. His father, Constantin, was extra- 
ordinarily brilliant; a poet, student of natural philosophy, 
classical scholar and diplomat, he typified the conception of 
culture at its best at the beginning of the century. As secretary 
to the Prince of Orange, Frederick Henry, he must be considered 
important in the guiding of the country through difficult times. 
In this, however, his own father, an earlier Christian, stood as an 
example, for he had been secretary to William the Silent in the 
eventful years after 1578. This Christian was a native of the 
Southern Low Countries, while his wife, Susanna Hoefnagel, was 
of Antwerp, though at the time of her marriage a protestant 
refugee from Amsterdam. The two sons, Maurice and 
Constantin, were born in troubled times, the latter on September 
4th, 1596, at the Hague. 

The last quarter of the sixteenth century saw the indepen- 
dance of the seven northern provinces of the Netherlands 
regained after an eighty years' struggle with Spanish power. In the 
South, Spain and Catholicism continued to dominate; in the 
North, religious and political liberation occurred together and 
there grew up a deep mistrust of all hierarchial powers; even the 
doctrines of Luther were rejected because they acknowledged the 
authority of the State in religion. A new Calvinist common- 
wealth now existed, and its rise has been described in the pages 
of Motley's Rise of the Dutch Republic. The assassination of 
William the Silent, in 1584, came after he had accomplished his 
great task for he had, as Motley says, " planted a free common- 
wealth under the very battery of the Inquisition in defiance of 
the most powerful empire existing ". 

It is interesting to look back at these important events which 
came close to the life of the Huygens family. When Maurice of 
Nassau was engaged in defeating the Spaniards in the open field, 
Constantin Huygens, father of the scientist, was receiving a care- 
ful and thorough education as a boy. This Huygens showed 
quite a distinction in mathematical work but all the influences 


of his life were in the direction of the courtier and diplomat. He 
was often at the court of Louise de Coligny, the widow of 
William the Silent, and he accordingly spoke French from boy- 
hood. He completed a course of Law at Leyden University and 
then was introduced at twenty-one to the life of diplomacy. This 
Huygens became by far the most well-known member of the 
family up to the mid-seventeenth century. His all-round culture 
has been mentioned, and he did in fact become known all over 
Holland, and in England, as a latinist and poet, as an amateur 
of music and painting, and as a student of philosophy. He was, 
besides, a close friend of Descartes and, at length, best known 
of the leaders of contemporary thought in the Republic : " no 
Dutchman commanded a more European culture; no Dutchman 
was more thoroughly Dutch ". After their first meeting, 
Descartes wrote of him " . . . despite what I heard of him, I 
could not believe that a single mind could occupy itself with so 
many things and acquit itself so well of them all ". 

Christian Huygens's father, then, was a man of outstanding 
ability and brilliance and he was very well known in England. 
He studied at Oxford for a time and became an intimate friend 
of John Donne. He played the lute at the court of James I, and 
in 1622 received an English knighthood. Nor was this brilliance 
a mere glitter, the effect produced by a versatile and fashionable 
courtier. Constantin Huygens corresponded for years with 
Descartes, with Mersenne, the great intermediary of men of 
science of that time, with Diodati, a friend of Galileo, and with 
many well-known mathematicians, notably Schooten the elder. 
In his MSS. have been found notes on Euclid's propositions and 
records of his study of optics. When Golius succeeded Snell at 
Leyden, Constantin Huygens recommended him to apply him- 
self to optics. "The consequences of the law of refraction 
[formulated by Snell in 1621] have not been sufficiently studied 
by anyone," lie wrote. He himself is said to have attempted to 
grind lenses to the forms proposed by Descartes the surfaces 
being of elliptical or hyperbolic section instead of spherical. 
Descartes had concluded that such lenses would be free from 
spherical aberration but Huygens (or the skilled mechanic 
employed by him) found the work impossible with the ordinary 
tools then used. His indirect influence in scientific work was 
undoubtedly of greater significance : without his encouragement 
Descartes might never have published his Dioptrique. The 


philosopher was induced to overcome his well-known 
hesitancy only through the efforts of Constantin Huygens and 

This versatile man of letters and diplomat in 1627 married his 
cousin, Susanna van Baerle, daughter of a wealthy merchant of 
Amsterdam and by all accounts an intelligent and cultivated 
woman. The children of this marriage, which must interest 
geneticists, were Constantin (1628), Christian (1629), Louis and 
Philip, the last of whom died young, and Susanna. In 1637, after 
only ten years of married life, the mother of this family herself 
died. Another cousin took over the care of the family, which 
removed to a newly built country house at Voorburg, close to the 
Hague. Here, when he had recovered from the death of his wife, 
Constantin received officers of the French army, French 
diplomats and men of letters. Here Descartes himself made 
occasional visits and remarked on the prowess of young Christian 
in mathematics, a study in which he complained he saw no great 

Descartes spent a good deal of time in Holland and did much 
of his more important work in the quiet of the country. Even in 
Holland, however, he did not feel sufficiently secure to bring out 
his treatise Le Monde and it was not until 1637 that his Discourse 
on Method appeared. But it is easy to imagine the great influence 
of Descartes on the intellectual family at Voorburg in those 
years just succeeding the publication of the famous Discourse. 
The work itself shows the appeal of Descartes's mode of argu- 
ment and, to a generation who read and sympathized with 
Campanula's Defence of Galileo, it must have seemed that 
Descartes was indeed the apostle of intellectual freedom. 
Campanella's tract, composed in a Neapolitan dungeon in 1616, 
was printed at Frankfurt in 1622, and during the next thirty 
years it was widely read by educated men all over Europe. Its 
courageous stand for freedom of enquiry and for the truth of 
the Copernican theory was a source of inspiration. For it is clear 
that a generation which could revere Galileo did so because their 
minds were alreadly partly prepared by earlier critics of Aristotle: 
Benedetti, Stevinus and others. In Campanella there was a 
vigour and boldness which recalled Giordano Bruno. Bruno and 
Campanella held that there are an infinite number of worlds, and 
if in Descartes's writings this doctrine as well as that of 
Copernicus was taught with great caution there can be no doubt 


that this was through circumspection. Descartes was a cautious 
man but very probably in conversation he was bolder. 

Constantin Huygens was extremely proud of his two eldest 
sons, who early showed intellectual brilliance. They were taught 
at home by a private tutor until Christian was sixteen. This 
education included singing, playing the lute, and the composi- 
tion of Latin verses. Like Newton, as a young boy Christian 
loved drawing and the making of mechanical models on which 
he spent much labour and ingenuity. So much so that his tutor 
felt misgivings; such practical work was after all an inferior and 
even a dubious sort of occupation for a young man of family and 
position. From the beginning, however, Christian showed 
promise of great skill in geometry while his brother, Constantin, 
excelled rather in literary compositions. Descartes was much 
impressed with some very early work of Christian's and he saw 
that great things might be expected from this rather serious boy 
with the rather pale face and the large dark eyes. Christian was 
rather delicate and by nature gentle, and his sensitivity seemed 
almost feminine to his father, who seems to have been fortunate 
in possessing an unflagging and exuberant vigour, quite different 
in character from his son. 

Characteristically enough, the first experiments of the youth- 
ful Christian were in mathematics, and this is typical of him, for 
he rarely ventured publications on other than abstract and some- 
what theoretical subjects. But the influence of a cultured and 
enlightened society remained with him, and his interests, early 
determined, lasted unchanged all his life. 

In 1645, when he was sixteen, Christian and his brother 
entered the University of Leyden. Here they studied Mathe- 
matics as well as Law, the younger Schooten, a protege of 
Descartes's, then being professor. Schooten was an able 
mathematician and Christian acquired the reputation of being 
his best pupil. Mathematics was a subject which included what 
we would now call mechanics and, for example, in Stevin's 
Hypomnemata, a work in six volumes, there are discussions of 
centres of gravity, levers, simple machines and hydrostatics. 
Christian's father was clear about the supreme importance of 
mathematical training. In 1644 Descartes had published his 
Principia, a bold attempt to reduce all the changes of Nature to 
mechanistic processes and he, it was well known, exalted the 
study of the subject. Radical changes were taking place in men's 


ideas and during his time at Leyden Christian lived in an 
atmosphere of intellectual ferment. The ideas of Descartes were 
hotly contested by the Aristotelians and to such an extent that 
in 1646 and 1647 the university almost became a battlefield. Un- 
fortunately, there are only scanty records of Huygens's reactions 
to these experiences. Regarding Descartes's Principia he many 
years later remembered the deep impression it made on him. " It 
seemed to me when I first read this book, the Principia, the 
first time," he wrote, " that everything in the world became 
clearer and I was sure that when I found some difficulty that it 
was my fault that I did not understand this thought. I was then 
only fifteen or sixteen years old/' 

Descartes's ideas were strongly represented in Holland. Renier, 
one of his disciples, taught Cartesian philosophy at Leyden for a 
time and later went to Utrecht. Here 'he had great influence and 
was followed by Regius, one of his own pupils. Aristotelian 
philosophy was associated with the Jesuits and nowhere more 
than in northern Holland was their influence more strongly re- 
sisted. Nevertheless, even in Holland, freedom of thought was not 
absolute and only a few years previously the Aristotelians had 
scored notable victories by arousing suspicion as to the religious 
consequences of Descartes's teachings. Cartcsianism owed its wide 
appeal to the ned felt for a new celestial mechanics after the 
acceptance of the ideas of Copernicus. This, apparently, 
Descartes's theory supplied. Moreover, Aristotle's outlook in 
natural science was in the main teleological. It was felt that if the 
guiding principle of teleology were abandoned some way of 
expressing the determinism of events must be found. On this 
point Descartes's analysis proved less sound but his system as a 
whole was ingenious and even aesthetically satisfying. 

In 1647, after two years at Leyden, Christian Huygens joined 
his brother at the College at Breda. This college, founded by 
Frederick Henry, seems to 'have achieved a temporary fame but 
it came to an end during the century. Descartes seems to have 
taken some interest in the place and certainly the forces of Aris- 
totelianism were there unable to challenge the new philosophy. 
John Pell, an Englishman, taught mathematics and was a man 
of quite high reputation. It was fortunate that, after Schooten, 
Huygens had so able a teacher. 

As soon as Huygens's period at Breda was completed he made 
a number of journeys, first going to Denmark in the company of 


the Count of Nassau-Siegen and later, with Constantin, to Frisia, 
Spa and Rome. When in Denmark it was a great disappointment 
to him that the weather made it impossible to reach Stockholm 
for Descartes was then living at the court of Queen Christina. 

Travel and a thorough education were, however, not the only 
elements which made up the pattern of Christian's early years. 
Most important, perhaps, of all was the correspondence he took 
up with Pere Mersenne, who was next in importance to Des- 
cartes among his father's acquaintance in the centre of the 
learned world. Duhem has described Mersenne as a man of in- 
satiable curiosity and the exuberant imagination of the artist. 
He was at this time the great intermediary for scientific com- 
munications between the chief centres of experiment. He popu- 
larized much of Galileo's work and did much to thrash out those 
fundamental notions on which seventeenth century mechanics 
was based. Men like Descartes, Gassendi, Fermat and Pascal met 
together at the cell of the Minorite father in Paris and this group 
has been described as the origin of the Academic Royale des 
Sciences. Mersenne was indeed a remarkable man, for he retained 
the esteem of both Church and the scientific world ; "... he 
did not believe all his religion," Pineau wrote to Rivet, " he was 
one of those who are glad enough to see church service done . . . 
he dared not often repeat his breviary for fear of spoiling his good 
Latin." He was not himself a great originator. Pascal possessed 
for mathematical and scientific work all the qualities which Mer- 
sennet lacked: a profound penetration, logical rigour, critical 
acuity, but Mersenne saw clearly which problems then mattered 
most and Huygens was indebted to him for many of the subjects 
of his early researches. 

Aristotle, whose mechanics was the weakest part of his natural 
science, had supposed that heavy bodies fall towards the centre of 
the earth because this is their " natural " place. The heavier a 
body is, the faster it moves towards the earth. If it were to fall 
through a hole passing through the centre of the earth it would 
come to a standstill on reaching the centre. As early as 1585 Bene- 
detti had protested against this. He saw, in a general way, that 
the inertia of the mass would carry it past the midpoint and that 
it would in fact oscillate after the manner of a pendulum bob. 
Stevin, with greater certainty than in the case of Galileo, is 
known to have experimented by dropping large and small 
weights simultaneously and showing that they reached the 


ground together. Galileo made a more thorough examination 
of naturally accelerated motion and calculated the distances tra- 
velled in successive seconds by a freely falling body. Mersenne, 
in an early letter to Huygens, questioned if in fact the mass did 
not in some way determine the limit of the velocity which could 
be imparted. Huygens explained that his objections were all 
based on observations of air resistance and gave such an able ex- 
position of what is now termed Newton's first law of motion that 
Mersenne gave him ungrudging praise : " I assure you that I 
think so highly of your demonstration concerning falling bodies 
that I believe Galileo would have been delighted to have you as 
his follower/' Mersenne went on to set Huygens the problem of 
finding the form taken up by a rope hanging from its two ends 
which are fixed at the same height and some distance apart. Huy- 
gens did not solve this mathematical problem until he recurred 
to it late in life but he studied the disposition of weights along 
the rope which would give it a parabolic form. He also became 
interested in Mersenne's famous problem of determining the 
centre of percussion of suspended bodies. This most important 
problem was given its first general solution by Huygens many 
years later. 

Young Huygens was delighted with these letters, which he 
received " with joy and avidity ". His father noted with approval 
the penetration with which young Christian, then only seven- 
teen, tackled problems then exercising the world's foremost men 
of science. In December 1646 Christian wrote that he was occu- 
pied with problems of centres of gravity and with modern de- 
monstrations of some of Archimedes' propositions on the sphere 
and cylinder a remark which illuminates the nature of his early 
training " but nothing yet concerning centres of percussion of 
which you recently wrote. However, I shall not fail to do all that 
I can to find the demonstration although, up to the present, it 
seems to me to surpass my ability . . ." Mersenne acknowledged 
that he also could not see how a single rule could satisfy the 
variety of figures for which the centre of percussion (or of oscil- 
lation) was required. The problem was that of finding a formula 
which would make it possible to calculate for any suspended 
body the length of the simple pendulum which would have the 
same period of oscillation. An experimental solution could, of 
course, be found but this was not acceptable as an answer. It 
is at first sight surprising that a grert deal of interest should be 


aroused by so academic a problem. This was because the prob- 
lem was one concerning a dynamical system (as opposed to a 
single mass) and it was obvious that a new approach was needed. 
Problems of this sort led to the development of the calculus by 
Newton and Leibnitz. Huygens, however, obtained a solution 
in advance of either of them, although it was without the use 
of their modern methods. 

Some interesting matters are discussed in the correspondence 
of Mersenne and the elder Constantin Huygens. Christian, 
because of his precociousness, is sometimes referred to as a mod- 
ern " Archimedes ". Mersenne wrote about the new work of the 
young Pascal, then twenty-five, the problem of the nature of the 
vacuum, the development of the telescope and the most recent 
astronomical observations. There was a widely accepted belief 
that a true vacuum is contrary to nature and this made it very 
necessary to explain the well-known experimental results 
obtained by Viviani and Torricelli in 1643. 

The followers of Descartes were in obvious difficulties because 
Descartes rejected the atomic doctrine and with it the notion of 
a void. Since Gassendi was reviving, at least in part, the atomic 
doctrine of Epicurus and considered it a profound philosophical 
necessity that a vacuum should be possible, this apparently recon- 
dite and academic matter aroused vigorous controversy. Galileo 
attributed the more or less constant height of the barometer to 
an equilibrium between the weight of the column of liquid 
and an attractive " force " acting upwards. This force was, of 
course, quite an illusion. Torricelli and Viviani verified that the 
relative heights of liquid which could be supported in a baro- 
meter tube varied inversely with the densities and in 1644 Torri- 
celli really gave the correct explanation based on the pressure of 
the atmosphere. Four years later Pascal's explanation of the 
behaviour of the barometer was put forward, after the experiment 
carried out for him on the Puy-de-D6me in September 1648, but 
his views were by no means universally accepted. Quite a litera- 
ture was produced in disproof of the existence of the vacuum 
and even Constantin Huygens, who confessed himself most anxi- 
ous to " penetrate all the mystery ", found the new explanation 
very contrary to his inclinations, which were all for Cartesianism. 

It is important to realize the fascination which all but a few 
critical spirits found in Descartes's natural philosophy (see p. 109). 
Tn a sense, of course, Cartesianism was anti-scientific. At a time 


when the trend of natural philosophy was in the direction of 
empiricism, Descartes emphasized the great limitations of the 
empirical method. While he scorned scholastic logic he con- 
sidered that mere empiricism was futile and that his discovery of 
analytical geometry illustrated the true method by which physi- 
cal problems of all kinds involving motion in space could be 
attacked. He believed that the way was open to reduce all phe- 
nomena to the terms of geometrical description. From the pos- 
tulates of space and motion, without any assumptions as to the 
innate properties of matter, he hoped, by successive applications 
of his intuitive method to isolated problems, to build up an 
account of all the phenomena of the Cosmos. The a priorism of 
Descartes's method is thus anti-scientific. On the other hand it 
must be remembered that Aristotelian science was concerned with 
logical rather than spatial relations. Descartes, on the other hand, 
has been well described by the remark that he was " the author 
and prophet of mechanism". With Galileo he asserted the 
belief that the laws of Nature are both simple and open to dis- 
covery. The danger for a youthful student such as Huygens was 
that Descartes paid too little regard for what have been called 
" stubborn and irreducible facts " and that he strayed too far from 
the path of scientific work in undertaking to heal the schism 
between the natural and the revealed. 

In the famous vortex theory of the Principia Philosophize 
(1644) Descartes supposed all space to be filled with a " subtle 
matter " which moved with the planets in their paths. He made 
brilliant play with this medium and used it to work out plausible 
explanations of gravity and magnetism as well as the action of 
the barometer. Light was treated as an action or as an inclination 
to move, possessed by the particles of the subtle matter. From 
this explanation, comparable with the idea of pressure in a liquid 
or of impact amongst panicles in motion, Descartes attempted 
to derive the laws of reflection and refraction. The same spirit 
was shown in the mechanical explanations offered in the 
Meteors : atmospheric phenomena and the rainbow were given 
explanations based on known or partly known scientific prin- 
ciples. In the Principia Philosophise Descartes dealt, among other 
things, with the nature of matter and the general laws of motion. 
The whole of this "system" rested on insecure foundations and 
there was a temptation to ignore small but " stubborn and irre- 
ducible facts " which did not fit in. Since a perfect vacuum was 


something to which Descartes denied existence the Torricellian 
space in a barometer tube was supposed to be filled with the 
Cartesian subtle matter which, like the ether of the nineteenth 
century scientists, penetrated almost everywhere. A crucial ex- 
periment was performed in which a sealed and empty bladder was 
placed in an evacuated tube. The fact that the bladder expanded 
was explained by Huygens as due to a small amount of residual 
air: Roberval, an original and controversial writer, also con- 
sidered the experiment disproved rather than supported the 
Cartesian theory. 

Mersenne died in September 1648, but his influence on Huy- 
gens had been important. Although he was no great physicist or 
mathematician he stimulated the criticism of ideas; he was, for 
example, strongly opposed to Descartes's well-known treatment 
of animals as automata. In the years between 1648 and 1657 Huy- 
gens, from being a youthful admirer of Descartes's philosophy, 
became more and more critical. He wrote frequently to his old 
teacher Schooten and to the mathematician Slusius about Des- 
cartes 's demonstrably false laws of impact between elastic bodies. 
The laws, he wrote, did not agree with any experiments and the 
fifth law conflicted with the second. Before 1656 he had com- 
pleted his own important work on the subject (see p. 109) but 
some twelve years elapsed before he communicated his conclu- 
sions to contemporary men of science. The complete treatise, 
De Motu Corporum ex Percussione, was not published during his 

The first published work of Huygens came out, however, as 
early as 1651, when he was only twenty-two. This was his Cyclo- 
metrise, a treatise written to show up the fallacies of the mathe* 
matician Gregory de St. Vincent committed in a book of 1647 
where Gregory had claimed to have developed no less than four 
different ways of " squaring the circle ". The task of replying to 
Huygens's serious objections was left to certain pupils, and 
notably to Ainscom. The result was considerable prestige for 
Huygens, for he was seen to have proved his case. The larger 
work, De Circuit Magnitudine Inventa. which appeared in 1654, 
it is safe to say, assured him of a place amongst the leading 
mathematicians of the day. He was hailed as the reborn Vieta 
and compared with Pappus and Apollonius, two giants of 
classical Greek geometry. The comparison was, in fact, not inept. 
In the years following 1652 Huygens spent a lot of time on re- 


ducing to algebraic analysis problems which Archimedes, Nicho- 
medes, and other Greek mathematicians had been able to solve 
only through geometry. Without these early studies it may be 
doubted if Huygens could have succeeded in the great problems 
he was later to tackle. 

Before the death of Mersenne Christian had hopes of going 
to Paris in the company of his father, but the idea was post- 
poned. In 1649 came the first of two revolts by the nobility 
against the rule of Anne of Austria and Mazarin during the 
minority of Louis XIV. Until 1653 t " ie situation continued to be 
uncertain; twice Mazarin was a fugitive, Anne was hunted from 
Paris and the monarchy was in jeopardy. The rebellious nobility 
were in league with Spain and the times were not propitious for 
the Huygens' visit. Not until 1655 was the long-projected visit 
made. The intervening five years were spent chiefly on Huygens's 
early researches, interrupted only by another journey to Den- 
mark and some time spent in the Low Countries. Huygens's 
important work on telescope construction dates from these days. 
The first telescopes were made in Holland early in the century, 
but they were very imperfect and it is remarkable that observa- 
tions such as those of Galileo were ever made with such instru- 
ments. The task of improving the telescope occupied Huygens 
throughout his life and in this he was encouraged by his father 
and had, from time to time, the skilful collaboration of his 
elder brother. By means of his own telescopes Huygens 
made his important observations on Saturn and a copy of a letter 
written at this time bears two rough sketches, one of Jupiter and 
the other of Saturn showing appendages. The contents of the 
letter do not relate to these matters and the date of the draw- 
ings is uncertain. The discovery of Saturn's ring cannot be put 
earlier than February or March 1656. Before he went to Paris in 
July 1655, however, Huygens had made the interesting discovery 
of a satellite of Saturn. The study of this anomalous planet 
whose irregular contour was such a mystery was continued by 
his brother Constantin in his absence. 

The fertility of Huygens's mind at this period was truly 
astonishing and it can be matched only by comparison with 
Newton. Fundamental research in pure and applied mathema- 
tics, optical studies including important work on the theory of 
lens systems, the invention of an improved eye-piece for the 
astronomical telescope, and to crown the practical side of his 


work, the discovery of Titan, all belong to this period of 
his life. Yet there was a curious weakness in this energetic 
mind, a flaw implanted perhaps by Descartes's brilliant 
philosophizing. For the man of science who was himself 
a few months later to discover Saturn's ring seems to have 
concluded that, with the discovery of six planets and six satel- 
lites, the human mind had reached the limits of the solar system. 
This preoccupation with numbers reminds one of Kepler and 
shows how persistent were old currents of thought. 

In 1655 Louis XIV was only seventeen and France continued 
to be governed by Mazarin. The second "Fronde" was at an end 
but this disorder and scramble for power left its results, abortive 
though it had been. The conditions were precisely those which 
could not but impress Louis with the need of becoming master 
and his reign from 1661 onwards was characterized by the great- 
est absolutism. 

For a son of a noted Dutch diplomat and man of letters 
young Huygens remained extraordinarily aloof from the turmoil 
of events. His habits soon became those of a scholar and at 
twenty-six there was a marked vein of seriousness in his pursuits. 
During his five months in Paris, however, he revelled in the 
opportunities of pursuing the arts as well as the sciences. Music, 
the drama, and the society of intellectual and artistic people 
made life in the capital extremely interesting. At the country 
house of Conrart, the protestant secretary of the French 
Academy, he met Jean Chapelain, a mediocre but popular poet 
and a man of cultivated tastes, and Marie Perriquet, an attractive 
young woman who seems to have shown interest and some ability 
in scientific problems. The comic dramatist Scarron, the astron- 
omer Boulliau and the philosopher Gassendi were also among his 
new acquaintance. 

Gassendi was at this time an old man. It is not clear how 
much Huygens could have been directly influenced by the philo- 
sopher on the occasions when they met, but his indirect influence 
on Huygens and several other men of science was considerable. 
Gassendi was at that time an important, perhaps the most im- 
portant, opponent to Cartesian teachings. His objections, more- 
over, reflected the influence of Galileo: logical deductions for 


him were useful only so long as they did not conflict with the 
physical facts. For Descartes mathematical and logical deduc- 
tions could be valid irrespective of verification from experience. 
The fact that this appears to us an impossible and irresponsible 
attitude must be attributed to the influence of Gassendi, Huygens 
and Leibnitz as well as of Newton in the history of thought. 
Even in a more detailed way, however, Gassendi had an im- 
portant influence on Huygens. He held an atomic theory which 
was later developed by Boyle and with this went a belief in 
scientific materialism. To Descartes's " cogito ergo sum " he 
objected that existence might be inferred from any other action 
besides thinking. Unfortunately Gassendi's name has become 
linked, not only with the revival of atomic doctrines, but also 
with the doctrine of mechanism. As a matter of fact Gassendi 
did not take up this extreme position. The atoms of bodies, he 
held, were not eternal or unproduced or moving of their own 
accord a problem which he seems to have viewed with the same 
perplexity as we feel now for the rotation of nebulae. God, for 
Gassendi, was the creator and first cause, He was over and above 
the physical world. 

From 1653 up to his death in October 1655 Gassendi lived at 
the house of Habert de Montmor, a wealthy amateur of the 
sciences, who gathered together at his house, 7 rue Vieille du 
Temple, many who had formerly met at the cell of Mersenne. 
This " Montmorian Academy " was an important forerunner of 
the Academic Royale des Sciences. As in London and in Florence, 
an informal gathering of men, free from the " systems " of the 
universities, committed to no philosophy save that of enquiry, 
founded the modern organization of scientific work. Besides en- 
quiring into new phenomena something was done to conserve the 
past. Gassendi wrote a life of Tycho Brahe and Copernicus. It 
is worth noting that he preferred the cosmology of the former to 
that of the latter. His pupil, the poet Chapelain, took a keen 
interest in the progress of scientific studies and formed a strong 
friendship with young Huygens. Chapelain knew little science 
or mathematics but his zeal was great and he assisted Sorbi&re, 
permanent secretary to the Montmorian Academy, to draw up 
its rules and maintain its foreign correspondence. Chapelain's 
letters to Huygens, after the latter's return to the Hague, show 
that he kept his master's atomic doctrines to the fore and con- 
scientiously maintained a critical attitude towards Cartesianism. 


After Gassendi's death in 1655, in fact, his place in Chape- 
Iain's life was taken by the newcomer Huygens. Unfortunately, 
the death of Gassendi was the beginning of a series of troubles 
for Montmor. First he lost his child and then his wife fell ill; 
finally, when she was recovering, his sister died. These misfor- 
tunes brought to an end, temporarily, the meetings held at his 
house. There may have been other reasons, for Gassendi's assist- 
ant, de la Poterie, and Huygens disliked each other, and Pierre 
Petit and Thevenot were hostile to Sorbiere. Numerous petty 
squabbles occurred and marred the work of the " Academy " for 
a period. Nor were other societies in Paris more successful. 
Thevenot later started discussions at his own house (in 1663) but 
these ended in 1664 because of the expense, part of which was 
incurred by keeping the mathematician, Bernard Frenicle de 
Bessy, and the anatomist, Steno, at his house. Groups supported 
for a time by Henri Justel or the Abbe Bourdelot suffered no 
better. It was through experience of this sort, as will be seen, 
that leading amateurs of science were brought to the conclusion 
that the Government should be responsible for the maintenance 
of a permanent academy. 

At the time when Huygens first visited the Montmor group 
there was, as usual, little contact with the Sorbonne. Some of the 
university teachers attended meetings, but perhaps as much out 
of suspicion as out of sympathy. The ecclesiastical authority of 
the university colleges felt itself challenged by Cartesianism, and 
Gassendi's views were no more popular. Nevertheless, a rational- 
ist sect existed within the Church and this was not wholly 
opposed to new ideas in natural philosophy. Among its 
supporters the writings of Du Vair and Descartes were in- 
creasingly popular. But the support of neither the Jesuits nor 
their opponents the Jansenists was of any value for the cause 
of science. 

Huygens enjoyed the meetings of the Montmor group and 
was anxious to prolong his stay in Paris. From the mathemati- 
cians he learned of problems on probability which occupied Fer- 
mat and Pascal about this time. It is of interest that his own 
little treatise on the subject, written after returning to Holland, 
became a classic. Before he left France he told Boulliau and 
Chapelain of his discovery of Saturn's satellite and the latter 
urged on him the importance of publishing the details. Huygens 
obviously wished to settle the problem of the ring first; he was 


pleased at the discovery that already his telescopes were as good 
as any that were to be had in Paris. 


In 1610 Galileo made a number of important telescopic observ- 
ations. In January he found that Jupiter had four satellites; in 
July he made out the appearance of Saturn as consisting of 
"three spheres which almost touch each other, which never 
change their relative positions, and are arranged in a row along 
the zodiac so that the middle sphere is three times as large as 
the others ". In the same year he distinguished separate stars 
in the Milky Way and saw the phases of Venus. His work left 
the Copernican theory in a much stronger position but certain 
unresolved doubts still remained. This " tri-spherical " form of 
Saturn for example was something completely anomalous. Were 
these outer spheres a peculiar type of moon? On the Copernican 
theory it appeared probable that other planets besides Jupiter 
would be found to possess satellites. 

This is where Huygens's work commenced. At the age of 
twenty-six he made a search for satellites of Venus and Mars, but 
in vain. Turning his own twelve-foot telescope on Saturn, it 
appeared to him much as it did to Galileo. The nature of the 
lateral bodies or appendages could not be distinguished. Leaving 
this problem on one side, however, Huygens at eight o'clock in 
the evening 'of March -25th, 1655, noticed a small star very near 
the line passing through the planet and its appendages or 
" anses ". His suspicions that this would prove to be a satellite 
were strengthened during the following days, for the position of 
the star altered. After a few weeks Huygens decided that the 
period of the satellite (Titan) was sixteen days and four hours. 

This discovery, as has been mentioned, was made before Huy- 
gens went to Paris in July. No doubt several of Montmor's group 
discussed with him the puzzling problem left by Galileo. Heve- 
lius, the noted astromer of Danzig, confirmed the planet's pecu- 
liarities, his telescope being not much better than that of Galileo. 
Huygens recognized that everything depended on improving the 
instrument. The diverging eyepiece of the Galilean telescope 
restricts the field of view and Scheiner, about 1630, successfully 
made the first instrument having two or three convex lenses. 
Telescopes of this type have a better field of view but are more 


subject to the defect of chromatic aberration, a matter not then 
understood. Huygens went to the lens maker Mocchi while he 
was in Paris and learnt all he could from him. When he returned 
to the Hague he worked continuously on lenses, trying amongst 
other things to produce a hyperbolic or elliptical surface, but both 
proved too difficult. He succeeded, however, in building a larger 
telescope having twice the magnifying power of his twelve-foot 
instrument and this enabled him to study Saturn more closely. 

In the winter months of 165556 great progress was made to- 
wards solving the problem. Instead of the " tri-spherical " form 
he was able to distinguish a sort of band passing across the 
middle of the planet and drew it in the form : 

A slightly later drawing of Saturn showed it in the form : 

It is difficult to imagine what his conjectures were at this point. 
A new twenty-three foot telescope with the best lenses he could 
make was assembled with all speed and this was in use after Feb- 
ruary ipth, 1656. With this instrument the planet appeared much 
more distinctly and he at last made a drawing of it showing it 
surrounded by a ring : 

The drawings he made show that he was struggling to gain a 
clearer image and only by degrees became certain about what he 
could see (cf. p. 194). 

During an interval between June and October the planet was 


not clearly visible but Huygens already felt confident that 
his observations had only one interpretation: Saturn is sur- 
rounded by a thin ring of matter slightly inclined to the ecliptic. 
This idea was concealed in an anagram published in his De 
Satitrni luna abservatio nova, which came out in the spring of 
1656. When disentangled the anagram reads "Annulo cingitur, 
tenui, piano, nusquam cohaerente, ad eclipticam inclinato," viz. : 
" It is encircled by a ring, thin, plane, nowhere attached, inclined 
to the ecliptic." From his correspondence it is clear that he was 
confident of his conclusions as early as February of that year. 

The use of anagrams was common in those days. Huygens 
adopted the device so as to give an opportunity for other astron- 
omers to bring their own discoveries to the light of day " so that 
it may not be said that another has borrowed from us, or we 
from him ". The method was superseded with the growth of 
scientific periodicals. In this instance, Roberval, Hevelius and 
one Hodierna all came forward with their own announcements. 
Hevelius alone produced anything of importance. His Disser- 
tatio de Natura Saturna Facie (1656) contained in fact a complete 
theory for the observed periodicity in the phases of Saturn. He 
cannot have seen the planet clearly for he supposed it to be ellip- 
soidal and not spherical in form; also there were two appendages 
physically attached to its surface. 

Roberval put forward the theory that Saturn is surrounded by 
a " torrid " zone. From this equatorial zone " exhalations " were 
ejected and these were supposed to be transparent except when 
present in great quantities. The periodicity in the phases was 
ignored. Even more remarkable was Hodierna's account of 
Saturn. His theory that the planet had the form of an egg or 
plum having two dark patches deserved careful verification, Huy- 
gens caustically remarked. Certainly such an appearance called 
for study by a better telescope than one of a magnification of 

Bouliiau was unable to see the satellite Titan and this made 
Huygens suspect the quality of his telescope and for this reason 
to trouble little about Boulliau's criticism of the ring theory. It 
mattered rather more when Wallis, the English mathematician, 
wrote to say that the English had forestalled him. This, however, 
proved to be a practical joke by Wallis, who otherwise is known 
only for the seriousness of his pursuits. 1 However, the details 

1 See notes on Wallis and others (p. 212). 


Saturn Reproduced from Huygens's MS. 


of the ring continued to give Huygens, as he said, " no little 
trouble ". The difficulty was to fix the interval between the phases 
and calculate the future appearance of the planet. At the end of 
1657 Huygens was able to inform Boulliau of the confirmation 
of his theory. " On the ifth of December I saw Saturn with my 
big telescope for the first time after it had passed the sun and was 
delighted to find it exactly in the form I had predicted according 
to my hypothesis." He went on to say that the ring appeared a 
good deal larger since its last occultation " so that now the sky 
can be seen through it ". 

At the crowded assembly of Mommor's circle Chapelain pre- 
sented a detailed account of Huygens's studies of Saturn. The 
planet was in all other respects normal: it traversed an orbit 
around the sun and its axis of rotation was almost parallel with 
that of the earth. The axis was always perpendicular to the 
plane of the equatorial ring. The solid and permanent nature 
of the ring could be clearly perceived. Twice in thirty years 
the sidereal period of the orbit the ring appeared to vanish 
since it was viewed edge on. The company was a distinguished 
one and general praise was forthcoming for the young astrono- 
mer's discovery. Even Roberval paid him a generous tribute 
and retracted an earlier suggestion that Huygens was indebted 
to him for his ideas. He still maintained, however, that his 
own theory was to be preferred. Huygens wrote that the ring 
was without doubt a great novelty and one to which " in the rest 
of the universe there appears to be no parallel ". In June 1659 his 
book Systema Saturnium appeared. 

Copies of the little treatise were sent to Paris and to Prince 
Leopold de Medici, to whom it was dedicated. The prince was 
a great supporter of science and founder of the Accademia del 
Cimento. Influenced by Boulliau, who sent him criticisms of 
the hypothesis, however, Leopold hesitated to express his opinion 
of Huygens's work and it was only after a considerable delay 
that he acknowledged its importance. Of the noted astronomers 
of the day, Hevelius, Boulliau and Riccioli did not accept Huy- 
gens's account of Saturn's ring. To this day no-one seems to have 
recognized the importance of Huygens's theory that the ring 
would be stable under uniform gravitational attraction assum- 
ing mechanical resistance to fracture. He did not state that the 
gravitational force kept the ring in rotational stability but he 
did suggest that Saturn's gravity extended to the ring. 



It was as a comment on Copernicanism that Huygens intended 
his book to be read. The nature of gravity was, he insisted, the 
same for all the planets. A stationary ring of uniform thickness 
would, then, be in equilibrium. No further evidence could well 
be expected. There are other interesting matters in the work, 
but these will be discussed later. What concerns us here is that 
the severest attacks which were made on the Systema Saturnium 
were made for religious and not scientific reasons. It is curious 
that after a period of tolerance the Catholic Church became 
bitterly opposed to Copernicanism in the seventeenth century. In 
1615 the Holy Congregation had declared all books of Coper- 
nican doctrine to be condemned and prohibited. Chapelain cer- 
tainly expected trouble and wrote that it was surprising that the 
hypothesis of the movement of the earth was allowed to pass in 
Holland. Pre Honori Fabri, a Jesuit, and an astronomer, 
Eustachio Divinis, were foremost in their antagonism to Huy- 
gens. These critics found it necessary to impugn not the argu- 
ments advanced by Huygens but his very observations. This 
drew a sharp reply from Huygens in his tract Antidivmis, but 
the controversy dragged on until Huygens was well established 
in Paris in 1666 and will need a further account later. The name 
of Fabri is obscure enough now. Nevertheless, under the name 
of his friend and pupil, Mousnier, appeared one or two inter- 
esting attempts to develop mechanics. The trouble was that 
Fabri possessed the outlook of an Aristotelian, for he wished to 
deduce the mathematical laws of dynamics from principles of 
natural philosophy. So deeply rooted was this habit of mind 
at this time that considerable feeling was frequently roused by 
ideas which rested on an entirely different attitude. In France 
the authorities of the theological college of Paris University tried 
to get decrees issued in defence of Aristotle's philosophy and 
against the new heresies as late as 167 1. This ridiculous situation 
was treated to a sarcastic burlesque by the playwright Boileau, 
who thereby did much to wreck the scheme. Later, a more eclectic 
outlook existed; Cartesianism entered the Sorbonne itself. 

Between 1655 and 1660 Huygens spent much time on the 
invention of an accurate pendulum clock. The significance of 


this invention in the history of science is that it marked the new 
interest taken in time as a dimension. We shall see that Huygens 
effectually began the study of dynamics. One reason why the 
history of mechanics up to his time was really the study of 
statics was undoubtedly the tendency resulting from the neo- 
Platonic revival of the sixteenth century which coincided with 
the decline in Aristotelianism. This tendency was to reduce most 
physical problems to geometry. But the absence of accurate time 
measurement was undoubtedly another reason. Galileo, it will 
be remembered, used a water clock in his experiments on acceler- 
ation over the inclined plane. 

Very probably it was in the first place Huygens's early enthu- 
siasm for astronomy which led him to tackle the problem of the 
pendulum clock. Balance clocks existed, of course, from much 
earlier times, probably from the thirteenth century, but they 
were crude and unreliable machines. Tycho Brahe used one in 
conjunction with his mural quadrant and corrected for its errors 
by comparison with the sun. The measurement of the time of 
passage of a star across the meridian could be used to replace the 
measurement of its meridian altitude, this being a more difficult 
measurement and rendered uncertain through the absence of 
reliable corrections for the atmospheric refraction. Also, as a 
member of a seafaring nation, Huygens could not have failed 
to know that an accurate clock would afford the simplest method 
of determining longitudes at sea. This question seems to have 
interested him more after he had made his first clock. 

His first publication, Horologium, a short treatise describing 
the application of the pendulum to the escapement, appeared 
in 1658 but the invention was known to his friends some two 
years earlier. Unfortunately a controversy arose through the 
claim made by Leopold de Medici that the priority for the in- 
vention belonged to Galileo. Roberval and a Paris clockmaker, 
Thuret, also claimed that they had anticipated Huygens. The 
whole history of the pendulum clock has in fact been obscured 
by various energetic contestants. 

In 1598 the King of Spain offered a prize of one thousand 
crowns for a means of finding longitudes at sea and this was 
followed by an offer of ten thousand florins by the States General 
of the Netherlands. Now Galileo is said to have discovered the 
approximate isochronism of a simple pendulum in 158 1 . He him- 
self, in 1636, offered to the States General a method of determin- 


ing longitudes based on the telescopic observation of the 
occultations of the moons of Jupiter. It was proposed to publish 
an almanack of the eclipses of these moons and to use a 
" numeratore del tempo " to measure the time intervals. This 
instrument, from all accounts, was merely a simple pendulum 
maintained swinging by hand and fitted with a completely im- 
practicable mechanism for counting the swings. Admiral Read's 
committee did well to reject the " invention ". It is quite possible 
that in 1637 Galileo came across Leonardo da Vinci's drawing 
for a clock regulated by a pendulum. It was in this year that 
da Vinci's manuscripts were given to the Biblioteca Ambrosiana 
at Milan by Galeas Arconati and the donor is known to have 
been at pains to bring his treasure to the notice of contemporary 
men of science. Galileo in this year became blind after a long 
period in which his eyes were diseased but he had around him 
Viviani, Torricelli and his son Vincenzio. Viviani, writing to Leo- 
pold in 1659, described from memory how Galileo discussed with 
his son the construction of a pendulum clock. The date given 
was 1641. Whether Vincenzio ever completed its construction 
is not known. It is certain that Huygens knew nothing about 
the design until after the publication of his Horologium in 
1658. A copy of this was sent to Leopold de Mediei, who replied 
guardedly, pointing out that Galileo had had the same idea. 

As against the theory that Galileo was indebted in any way 
to da Vinci it needs to be mentioned that the design commonly 
attributed to Galileo differs somewhat from that shown (though 
rather imperfectly) in da Vinci's note-books. Huygens's design 
differs from that of Galileo and was, in fact, closer in prin- 
ciple to that of da Vinci. 

It was in the records of the Accademia del Cimento of 1662 
(published in 1667) that the implication of Huygens's plagiarism 
was really blazoned abroad. Here it was stated that Vincenzio 
had put his father's design into practice in 1649. No details were 
given and the illustration simply showed a drum-shaped clock 
mounted horizontally on a vertical pedestal. From the under- 
neath side of the drum hung something resembling a simple 
pendulum, the mode of attachment of which had to be 
imagined. It is obvious that a simple pendulum would be useless, 
since it is impossible to give an impulse to the thread and pre- 
sumably a thin iron rod was intended. Nevertheless, Matteo 
Campani stated that he saw the clock constructed by Vincenzio 


" an antique and rusty machine not at all complete " and 
Leopold's letters certainly suggest that such a clock was in 
existence, though whether it conformed to the diagram is not 
known. The clock was never forthcoming and it has generally 
been supposed that Viviani pressed the whole case for Galileo out 
of a desire to honour his master. Nevertheless, the evidence does 
seem to indicate that Galileo did precede Huygens in achieving 
the successful application of the pendulum to the escapement, 
but that the complete clock was constructed is exceedingly doubt- 
ful. Probably da Vinci was the first to have the idea and still 
more probably Huygens was the first to carry it through to 
fruition. In a later chapter it will be shown how astonishingly 
thorough in every particular Huygens's work was; so completely 
did he clear up the theoretical and practical problems that he is 
in a real sense the father of modern time-measurement. Samuel 
Coster, his clock-maker at the Hague, made a large number of 
clocks to his design and these were the first to be commercially 

As has been mentioned, Huygens probably interested him- 
self in the longitude problem after he had made his first clock. 
He may have read the work Nieuwe Geographische Onder- 
wijsinge in Dutch by Metius (1614) which pointed out that it was 
only the irregularity of balance clocks which prevented them 
from supplying a means of finding longitudes at sea, but in any 
case the relation between local time, standard time and longitude 
was well known. The pendulum clock was a far better instrument 
but it was very easily disturbed. Huygens consistently under- 
estimated this problem of the movement of the ship. He 
thought that a pendulum whose period was independent of the 
amplitude of swing would enable this difficulty to be overcome. 
The idea was ingenious and led to the discovery that the period 
of oscillation of a cycloidal pendulum is independent of the 
amplitude, but the practical value of the discovery was strictly 
limited. A great deal of time was occupied by the investigation 
of this problem, conducted as the research was by elementary 
and tedious mathematical methods. In practice a cycloidal 
pendulum may be constructed by allowing a simple pendulum to 
swing between two curved metal plates along which the thread 
curves itself on each half of its swing. In applying the idea to 
the clock pendulum Huygens used a short ribbon attached to a 
rigid pendulum. These plates or " cheeks " were first tried in 


1657 or during the last days of 1656. Towards the end of 1659 
Huygens showed that theoretically they should themselves 
possess the form of cycloid arcs. He was immensely pleased 
with this discovery and ranked the geometrical part of 
the work above all the rest. 

About this time (1659) more detailed accounts of 
Galileo's escapement became available. Models of this 
escapement have since been made and it cannot be 
said to work satisfactorily. As Huygens pointed out 
at the time this escapement imparts a very uneven 
movement to the pendulum. His own clock remained 
therefore the only one in this field. 

Continued efforts by other inventors, including the 
clock-maker Thuret, to profit from the invention 
drove him to the unwelcome decision that he should 
take out a patent or " privilege " to protect his rights. 
Much delay occurred before the French " privilege " 
was issued, but thereafter Huygens's priority was 
recognized and he made some profit from the construction of 
clocks to his design. 

The story of the clock needs to be told with reference to 
certain of Huygens's mathematical researches. His first essay in 
this field had dealt (1651) with some fallacious work by Gregory 
de Saint Vincent on the rectification (or measurement) of certain 
curved lines. Huygens became interested in the rectification of 
curved lines known as conies and in the age-long problem of the 
rectification of the circle. When Boulliau sent him some problems 
by Pascal on the curve known as the cycloid in 1658, therefore, 
the subject was by no means a new one. These problems, to which 
Pascal had already obtained solutions, and which he set for the 
interest or exasperation of other mathematicians, were known 
as the " Dettonville " problems, this being the pseudonym under 
which they were issued. Huygens succeeded in solving some of 
the necessary preliminary problems but found the main ones so 
difficult that he declared himself unconvinced that they had ever 
been solved. When he later came across a rectification of the 
cycloid by Christopher Wren he expressed his admiration. It 
was, he commented, the first curved line known to be rectified, 
and he wondered if it were the only one which could be rectified. 
There was some correspondence between Huygens and Pascal 
on the " Dettonville " problems. Pascal praised Huygens's 





penduhim clock very highly, but such was Huygens's esteem for 
Pascal as a mathematician that he deprecated such mechanical 
inventions. There was, he remarked, little science or subtlety in 
such things. About this time Pascal's adherence to the Jansenist 
sect was somewhat weakened but his periods of religious pre- 
occupation invariably interrupted his most interesting work 
and his most promising friendships. So it was in his relations with 
Huygens. The latter was eager to collaborate but closer relations 
were frustrated. 

It is clear that at some time between Septetnber 1659 anc * 
January 1660 Huygens discovered the theoretical form of the 
small plates or " cheeks " for his corrected pendulum. These 
dates may be fixed by an examination of his correspondence. 
From this it seems that he did not at first use the metal plates 
except for clocks in which a large swing was employed. Later, 
taking the view that a marine clock would benefit from having a 
pendulum swinging through a large arc, he felt that it was im- 
perative to discover the theoretical form of the restraining plates. 
His success in this problem gave him the pleasure of a mathe- 
matician with a pretty solution. He announced that the second 
edition of his Horologium would contain "a fine invention 
which I added to the clock a little while ago ". It appears prob- 
able that, although Pascal's problems were of a very different 
character, the interest of the cycloid led Huygens to make his 

The improved clock was in use towards the end of 1669 and 
was adopted after that date as being the best time measurer then 
made. Earlier astronomers, notably the Landgrave of Hesse 
(who used balance clocks made by Byrgius in the sixteenth 
century), Tycho Brahe, and later, Hevelius and Mouton, recog- 
nized the importance of time measurements, but Roemer 
and Flamsteed, late in the seventeenth century, were really the 
first to use the clock systematically. Delambre, in his great 
Histoire de rAstronomie Moderne, states that Huygens " started 
the great revolution " in practical astronomy by the invention of 
the pendulum clock. 

Curiously enough the cycloidal pendulum, in spite of its 
elegance, did not have a very long life. It was recognized by 
Huygens himself that small circular arcs were equally accurate 
and in his model of 1658 he was able to restrict the size of swing. 
After the application of the anchor escapement by a London 


clock-maker, Clement, in 1680, few cycloidal pendulum clocks 
appear to have been made. The anchor escapement causes the 
pendulum to describe small arcs of constant amplitude and this 
made the cycloidal pendulum for all but marine clocks super- 

As for the marine clock or chronometer, although Huygens 
constantly considered himself near to success, it proved eventu- 
ally to be a failure. The pendulum seemed for many years to be 
the only means of controlling the going of the clock with 
sufficient accuracy. Huygens accordingly tried various forms of 
suspension and various forms of pendulum, all designed to with- 
stand the movement of the ship, but none proved to be a practical 
proposition. Such was the commercial rivalry of the various East 
India Companies, however, that he was encouraged to persevere, 
and persevere he did up to the last year of his life. How near 
he came to success will be described later. 

Meanwhile the astronomers obtained their longitudes by 
Galileo's method of observing the recurrent eclipses of the 
satellites of Jupiter. Cassini, working at Paris, drew up the first 
tables for the observation of these satellites and, with Richer, in 
consequence of this work was able to make the first modern 
estimate of the distance of Mars. But astronomical methods 
were clearly unsuited to the determination of longitudes at 

Huygens's great essay, Horologium Oscillatorium, on the 
construction of the clock and all the relevant propositions on the 
cycloidal pendulum and on the centre of oscillation did not 
appear until 1673. Already, however, he had made notes for the 
work and even wrote to Chapelain in September 1660: "The 
treatise on the clock has been finished a long time but tnere is 
no means of having it printed before my journey ..." This 
refers to an extended edition of the original Horologium which 
included a treatment of the cycloid; much was yet to be added 
before the work reached its final form. In October Huygens left 
the Hague for Paris. 


In considering the encouragement given to literature and the 
arts in France during the seventeenth century, the credit belongs 
almost as much to Mazarin as it does to his successor Colbert. 


Mazarin it was who gave pensions to many of the great writers 
who made this the golden age of French literature. Molifcre, 
Balzac, Descartes, Pascal, Racine, Corneille, Boileau and others 
benefited from Mazarin's patronage and his example was 
followed by Colbert after 1661. 

Nevertheless, the establishment of the Academic Royale des 
Sciences would never have been achieved if the men of science 
had waited for Colbert. Up to 1663 what progress there was in 
any regular pursuit of science was made by men who were 
associated with one or other of the amateur societies, Montmor's 
and Thevenot's being much the most important. But in this 
year there was a rather defeatist air about the correspondence on 
the subject of a permanent academy and Sorbiere summarized 
the difficulties in an account sent to Colbert. There had to be 
appeals by Sorbtere, Thevenot and the Abbe d'Aubignac, how- 
ever, before any impression was made. Finally Auzout publicly 
appealed to the king's pride (and vanity) and after the Peace of 
the Pyrenees things began to look more hopeful. 

The scientific societies did indeed develop under difficult con- 
ditions. In England, for example, where the Royal Society was 
taking shape, there was in progress a stern struggle between king 
and parliament, a deep religious dissension and in the com- 
mercial sphere a rivalry with the Dutch which had become acute. 
In Holland, on the other hand, it was understood that the chief 
political problem of the time was the neutralizing of the grow- 
ing power of Louis XIV. Civilization was passing through a 
critical period and internal dissension as much as external danger 
made the times, one would have thought, unpropitious in most 
countries of Europe for the growth of societies with the calm in- 
terests of natural science as their pursuit. But perhaps these 
interests were all the greater attraction; as Sprat wrote after- 
wards, the members of the Royal Society wished simply for " the 
satisfaction of breathing a freer air, and of conversing in quiet 
with one another, without being engaged in the passions and 
madness of that dismal age ". Needless to say their work was not 
always taken seriously and both in Paris and London there were 
scoffers who doubted the worth if they did not mistrust the in- 
fluence of " natural philosophy ". Pepys recorded that Charles II 
" mightily laughed at Gresham College for spending time only 
in weighing of ayre and doing nothing else since they sat ". But 
Colbert certainly saw that there was more to those pursuits than 


met the eye. It may be doubted if Louis XIV, unaided, saw any- 
thing significant at all in what was going on. 

In spite of the difficulties an extensive correspondence was 
carried on between the men of science. Paris was at first the 
chief centre of experiment, but London later rivalled and then 
surpassed it in activity. The meetings of the scientists seem to 
have been devoid of political motive; religious difference only 
rarely caused antipathy, national differences scarcely ever. As 
will be seen, Huygens spent many years in Paris under the 
patronage of Louis XIV, even though his family had a long 
association with the house of Orange and, after 1672, the young 
Prince of Orange headed the resistance to French invasion. It 
might have been thought that Huygens, a protestant Dutchman, 
would have been regarded as a spy but this was not the case. Not 
until 1683 did it become clear that, with Colbert's death, support 
for his continuance in Paris was gone. Colbert was succeeded by 
Louvois, and in 1685 the Revocation of the Edict of Nantes 
caused many protestants to leave the country. All this, however, 
lay in the future. 

In 1658 Montmor charged Sorbiere with the task of drawing 
up rules for the meetings of the assembly which were held 
regularly at his house. The keynote of the rules, in the form 
finally adopted, was the need of restricting " the vain exercise of 
the mind in useless subtleties ". Mere philosophizing, it was 
agreed, was profitless. Unfortunately the assembly did not 
appreciate how their aim should be achieved : without a pro- 
gramme of experimental work directed to the solving of selected 
problems too many of their meetings continued to degenerate 
into philosophical combats. 

Huygens returned to this gathering of scientific amateurs in 
1660 and was introduced by Chapelain on November and. At 
Montmor's, he wrote to his brother, " there is a meeting every 
Tuesday where twenty or thirty illustrious men are found 
together. I never fail to go ... I have also been occasionally to 
the house of M. Rohault, who expounds the philosophy of M. 
Descartes and does very fine experiments with good reasoning 
on them ..." Rohault's meetings were held on Wednesdays 
and began about 1658. There is no doubt that he did a great deal 
to make science popular in Paris. Unauthorized editions of his 
lectures were published, so great was the popular interest. It is 
not surprising that Huygens approved of him, although no 


association between them seems to have occurred. Rohault was 
enlightened and modern in his attitude but he was an expositor 
and lecturer rather than an original thinker. Where he abandoned 
Aristotle he followed Descartes. 

At Montmor's house Huygens noted a room "full of 
beautiful paintings ", a cabinet of curious inventions and mathe- 
matical instruments, and drawings by Albert Durer. With the 
astronomers he discussed his work on Saturn and the problems 
of lens grinding; with the mathematicians, as he noted, " my 
theories of the superficies of conoids and spheroids and the new 
properties of the cycloid for pendulums "; and with the clock- 
makers and telescope-makers Huygens also passed interesting 
hours. He met Conrart, Roberval, de Carcavy, Pascal, Pierre Petit, 
Sorbiere, Desargues and others. Some of these names will recur 
later. 1 He corresponded with Fermat and with his friend Boulliau, 
then staying with Hevelius at Dantzig. With Robert Moray, 
a prominent member of the London group of " scientists ", 
he also began a correspondence. These men were all really 
amateurs and the title astronomer, in most cases, for example, 
simply indicates the kind of work for which a particular man 
showed especial interest and in which he spent his leisure time. 
Nevertheless, a man like Cassini, later invited to work at Paris, 
represents the new type of professional worker in that most of his 
time was in fact spent on genuine systematic work. Huygens 
also belongs to this class. Real specialization in the modern sense 
was quite unknown, of course. Like all the early scientific assem- 
blies of the mid-seventeenth century the Montmorian society 
cast its net almost too wide. Huygens's diary records dissections 
of human bodies, the examination of machines for which per- 
petual motion was claimed, the making of lenses and telescopes 
and many other matters. Too much time, Huygens considered, 
was spent in arguments of a purely philosophical nature. He 
felt that a sterner discipline, a greater application, was needed 
than could come out of the performance of merely curious ex- 
periments and the holding of discussions. 

Nevertheless it must have been an interesting and stimulat- 
ing experience to have met so many natural philosophers, all of 
whom felt the common interest in the new study of nature. At 
the house of the Due de Roannes, Huygens, in December 1660, 
met Pascal. Eight days later the Duke, with Pascal, visited 

1 Sec notes on Persons Mentioned, pp. 212-6. 


Huygens at his lodgings in rue Sainte Marguerite and, wrote 
Huygens, "... we talked of the force of water rarefied in 
cannons and of flying; I showed them my telescopes." Pascal was 
at this time a sick man. The writer of Provincial Letters had, in 
fact, by this date retired more or less completely from the world. 
Less than a year later he was dead. 

The publication at this time of tracts against Huygens's 
account of Saturn shows that the orthodox Jesuits were not pre- 
pared to ignore the author's Copernican doctrines. Pere Fabri 
especially opposed what he called Huygens's "furtive insinua- 
tion" of the Copernican "error". In his Brevis Annotatio in 
Sy sterna Saturnium Christiani Eugenii (1660) he presented his 
own fantastic theory, although this work was published over the 
name of the astronomer Divinis. According to this theory the 
planet had two luminous bodies (lucidi) and three dark ones 
(obscuri) placed around it and the different relative positions of 
these bodies were the cause of its observed phases. This criticism 
drove Huygens to compose his Brevis Assertio Systematis 
Saturnii within the year. Hevelius is said to have been so favour- 
able to this reply that he abandoned his own theory in favour of 
the theory of the ring. Leopold, to whom both Divinis and 
Huygens dedicated their publications, remained reserved. In 
1 66 1 he sent Huygens a further pamphlet by Divinis and Fabri 
but it did not seem to Huygens to deserve a reply. It is note- 
worthy that by January 1665 even Fabri recognized the truth of 
the ring theory, convinced at last by the excellent telescopes of 
Guiseppe Campani. Huygens was extremely pleased by this 
conversion of his critic. "No-one, 5n my opinion, could reason- 
ably reproach me " he wrote, " for having adapted my account 
of Saturn to the system of Copernicus . . . the truth of the 
matter can only be explained by following Copernicus, and 
indeed our system of Saturn corroborates his own strongly." 


One of the reasons for Huygens's visit to London in 1661 was 
undoubtedly his desire to obtain information of the society of 
men of science then meeting at Gresham College, the society 
which by charter became in 1662 the Royal Society. He arrived 
in London in March just before the coronation of Charles II 
and left for the Hague at the end of May. 


The London which Huygens saw in 1661 was the London 
which was largely swept away by the Fire and he was not at all 
favourably impressed with the condition of the town. All, he 
found, was in marked contrast with Paris : the smoke from the 
furnaces of the brewers, soap boilers and dyers; and the stench 
of narrow alleys innocent of drainage and sanitation. Even 
Gresham College had been rendered malodorous. Monk's sol- 
diers had for a year used it as a barracks and it was, Bishop 
Wren wrote to a member of the society, " in such a nasty con- 
dition, so defiled, and smells so infernal, that if you should now 
come to make use of your tube [telescope] , it would be like Dives 
looking out of hell into heaven." This was in 1658 or the " fatal 
year 1659 ". With the " wonderful pacific year 1660 " meetings 
of the " invisible college " as Boyle called it, recommenced. By 
1 66 1 the college was presumably cleaned up. Huygens, at any 
rate, had only admiration for the proceedings there. Brouncker, 
Moray, Oldenburg, Godard, Boyle, Wallis and many others were 
familiar figures at the meetings of the society and their activities 
seemed to him to surpass anything done in Paris. The observa- 
tion of stars was done in the garden of Whitehall Palace and 
there Huygens tried his own telescope lenses, sent over by his 
brother Constantin. These proved to be better than the English. 
The Duke and Duchess of York came out to observe the Moon 
and Saturn. 

Huygens's meeting with Wallis is of especial interest. This 
great mathematician showed in his Mechanica sive de Motu 
(1669-71) that he had much to contribute to mechanics. The his- 
torian Duhem has given the opinion that this work was " the 
most complete and the most systematic which had been written 
since the time of Stevin ". In his work Wallis generalized the 
idea of force which up to his day was used only in connection 
with gravity. Huygens's English was at this time not at all good 
but he saw that it would be most valuable to keep in communi- 
cation with Wallis, as indeed with others of the Gresham College 
group. This was the begining of a life-long association with the 
English men of science. It is striking that, in spite of the official 
position Huygens came to have in the Academic Royale des 
Sciences, in 1670, when he feared that he had only a short time 
to live, he made arrangements to entrust his papers not to the 
Paris society but to members of the Royal Society. Through 
Oldenburg, the indefatigable secretary of the Royal Society, Huy- 


gens was fortunately able to remain in close contact with the pro- 
gress of science in England. 

There is good evidence that the outlook characteristic of the 
English men of science was less complicated by the consider- 
ations of a priori philosophies than that of the Paris group. 
Francis Bacon has, probably, always influenced literary men more 
than he has the men of science, but there is no denying his 
importance. Bacon was no scientist and his scientific "method " 
was the literary man's conception of science. He never advanced 
as far as Descartes into scientific studies. He foresaw, " he cast 
forth brilliant intuitions "; ridiculing Aristotle's natural philo- 
sophy he pointed to experiment and observations as the only 
means of discovering truth : " Nature to be commanded must be 
obeyed." With this spirit the English men of science, neverthe- 
less, were thoroughly imbued. Their opposition to Hobbes 
illustrated thir belief in empiricism. Hobbes's dictum " Experi- 
ence concludeth nothing universally " appeared to them mere 
philosophic wind; his excursions into physical science the sort of 
thing against which their motto Nullius in Verba was later 
aimed. It has to be admitted, of course, that most of the Eng- 
lish men of science were shocked by Hobbes's acceptance of the 
Epicurean philosophy. This appeared to them to be an approach 
to Nature which was neither scientific nor pious. Although he 
was not explicit about it, it was the more empirical attitude of 
the English men of science which impressed Huygens so favour- 

Before leaving London Huygens took part in a determination 
of the comparative sizes of the ring and globe of Saturn. He was 
pleased to find that his account of the planet was accepted with 
admiration. Huygens also saw a transit of Mercury from Long 
Acre, using one of Reeve's excellent telescopes. 

After Huygens's return to the Hague his father and younger 
brother went on a diplomatic mission to Paris. The elder Con- 
stantin Huygens, a man of European repute, then made the 
acquaintance of some of his son's associates. Through him the 
Montmor group heard of Christian's latest experiments. The 
Dutch diplomat took the opportunity to present Louis XIV with 
one of his son's pendulum clocks. This gift was opportune, for it 
was at this time that Colbert was drawing up his schemes to 
excel all past achievements in making Paris the cultural capital 
of the world and Louis the pre-eminent monarch of the age. 


Louis, of course, became surrounded by an almost ridiculous cult 
which sought to elevate him above everyday existence, but even 
this had its merits ! The singling out of writers, poets and men of 
science was at any rate one of the better consequences. These were 
given rewards totalling many thousands of pounds and the inven- 
tor of the pendulum clock later came in for suitable appreciation. 
More important, however, was Huygens's subsequent invitation 
to Paris to assist in organizing a scientific society under royal 
patronage. There were delays in carrying out this project, which 
must be ranked high among Colbert's achievements, but in 1666 
the societies which had been associated with the names of 
Montmor, Thevenot and others received this formal recognition 
of the importance of their work. 


The early scientific societies exhibited an enthusiasm and 
universal interest which scarcely characterizes the professional 
societies into which they have developed. Specialization was vir- 
tually unknown and through Latin the members had a means of 
communication with foreign societies and with a learned world 
which had existed before the new studies had begun. It is not 
surprising that men like Huygens were acquainted with the 
works of some of the Greek writers, nor that it was the Greeks of 
the Alexandrian period that held the greatest attraction. Huy- 
gens was only following in the steps of Galileo when he studied 
the works of Archimedes, for they contained some of the funda- 
mental ideas used in mathematics and in statics. But it was clear 
that new ideas of a fundamental kind were needed in mechanics 
and it was equally necessary to clear away many plausible 
suppositions which had no basis in fact. 

Huygens saw clearly that such simple machines as the lever, 
the pulley, and the wheel and axle, although thfcy gave a 
mechanical advantage, could in no way increase the energy avail- 
able. Machines for flying, for propelling boats by means of 
springs connected with trains of gears, strengthened his convic- 
tion that their limitations resulted from a simple mathematical 
identity of some kind. But it was many years before this idea 
could be expressed satisfactorily. Desargues seems to have tried 
to evolve a proof that perpetual motion is impossible but, proof 
or no proof, the impossibility was accepted as axiomatic within 


the realm of mechanics by Huygens. The search for a form of 
perpetual motion did in mechanics the sort of work that in 
chemistry was produced by the search for the philosopher's 
stone. In 1659 a book entitled Mechanica Hydraulico-Pneumatica^ 
by the Jesuit Schottus, reached Huygens. It was partly about 
perpetual motion but it also described Guericke's invention of 
the simple vacuum pump. In 1661, while in England, Huygens 
saw experiments performed at Gresham College using Boyle's 
pump, which was a great improvement on that of Guericke. After 
reading Boyle's book, New Experiments Physico-Mechanicall 
touching the Spring of the Air (1660) he had a copy of Boyle's 
pump constructed in November 1661. It is clear from his corre- 
spondence that he repeated many of Boyle's experiments, observ- 
ing for himself the boiling of water under reduced pressure, the 
absence of propagation of sound and the expiration of small 
birds in a vacuum. An important original discovery made during 
this work was that of the tensile strength of liquids, an effect 
which at that time baffled explanation and which led Huygens 
to make far-reaching conclusions on the existence of a subtle fluid 
or ether which later came into his theory of light. 

The question constantly in view behind all work with 
vacuum pumps was whether a complete vacuum could really 
exist. Many scientists felt, with Hobbes, that empty space is " an 
imaginary space indeed ". A fundamental experiment was to fill 
a tube with water and invert it so that the open end was under 
water in an open vessel and then to place the apparatus in the 
receiver of the air pump. When the pressure was reduced the 
liquid fell inside the tube and with continued pumping was 
brought down to the level of the water in the vessel. With rather 
more difficulty the same result was obtained, approximately, 
using mercury in place of water. These results agreed well with 
Pascal's explanation of the barometer. However, the appearance 
of small air bubbles in water which was placed in the receiver 
of the pump raised serious doubts, for the descent of the water 
and mercury might be attributed to the dilatation of these 
bubbles and not to the evacuation of the upper space in the tube. 
Huygens, much to his astonishment, found that if air-free water 
was used no descent occurred. If a very small bubble of air was 
introduced the descent took place. Boyle's law, however, showed 
that the magnitude of the effect was far too great to be 
accounted for on the dilatation theory. Huygens's observation 


was confirmed in England, and, at Boyle's suggestion, the effect 
was obtained without the use of a pump at all Long barometer 
tubes of mercury were inverted and were found by Brouncker to 
give the effect if air bubbles were carefully excluded. A column 
of mercury 75 inches long failed to descend unless a minute air 
bubble was present and then the level fell to the normal position 
of about 30 inches. 

For several years no explanation of this effect satisfied Huy- 
gens. But in 1668 he concluded that there must be a subtle fluid 
capable of penetrating glass where the contact of the liquid is 
not complete and that the height of the barometer is due to the 
combined pressures of this fluid and air. Wallis pointed out that 
if the subtle fluid were capable of penetrating glass it would 
penetrate the Torricellian space also. It is surprising if Huy- 
gens did not see the force of this criticism. Although he saw 
an analogy with the cohesion of two wet glass plates he missed 
the true explanation, which is that films of moisture (such as 
exist on mercury and glass) have considerable tensile strength. 
In his Traite de la Lumiere, written in 1678 and published in 
1690, much was made of this ethereal fluid in explaining refrac- 
tion. In fact, very great importance must be attached to Huy- 
gens's experiments with the vacuum pump, for his conclusions 
profoundly affected his whole outlook. He became an admirer 
of Boyle, whom he supported against the criticism of Hobbes 
and Linus. The former, he saw, contributed nothing to natural 
philosophy; of the obscure ideas of Linus (Francis Hall) he 
thought just as little. Boyle's Skeptical Chymist was published 
in September 1661 and Oldenburg gave Huygens an account of 
its contents. Later Huygens received a copy of the book which 
he read with " great pleasure ". " It contains an infinity of use- 
ful and remarkable things," he commented, " and in my opinion 
it is worth twenty of these other books which are continually 
printed on the matters of Philosophy and Chemistry. This 
Carneades certainly speaks very truly, reasons acutely, and with- 
out doubt shows the true way to discover the truth of things. ..." 
In 1662 Huygens heard of Boyle's famous experiments on the 
alteration of the volume of a gas with the pressure and he 
read Boyle's retort to Hobbes and Linus (A Defence of the Doc- 
trine touching the Spring and Weight of the Air). The kinetic 
theory of the gaseous state which originates with this work was 
propounded by Hooke as well as by Boyle. Hooke spoke of his 


theory as Epicurean after the Greek philosopher who, with 
Democritus, expounded an atomic doctrine. With the resuscita- 
tion of this doctrine in Europe Gassendi had a good deal to do. 
Hooke went into greater detail. The particles of air, he sug- 
gested, have " much the shape of a watch spring, or a coyle of 
wire " which, having rotatory motion, sweeps out a " potential 
sphere ", the volume of which varies with the closeness of the 
adjacent particles. Huygens was somewhat uncertain if this 
theory was in accord with the fact that at high pressures air 
retains its fluidity. 

If Huygens gained an interest in experiments employing the 
air pump from his visit to London in 1661, the English philoso- 
phers gained for their part just as much although in a different 
direction. For Huygens, by 1661, had discovered the use of a 
particular axiom in mechanics which enabled him to solve 
problems which Wren, Wallis and others found especially 
difficult. This axiom is a simple one : the centre of gravity of a 
system of bodies cannot rise as a result of any motion of the 
bodies under gravity. Experiments on the ballistic pendulum, 
carried out in Huygens 's rooms in London, showed that he could 
by this means calculate the heights to which elastic pendulum 
bobs would ascend after collision. Another discovery which 
greatly intrigued English mathematicians was the theorem that 
oscillations of a body in cycloidal arc, occurring under gravity, 
are truly isochronous. At the time he was in London Huygens 
had not worked out a complete proof although his note-books 
show that the work was well advanced. Many mathematicians 
were consequently attracted to the problem in the hope of being 
first to provide a proof. Brouncker and Auzout both failed, the 
former ignominiously, in attempting this problem, which is 
difficult by the old geometrical methods but simple when treated 
by means of the differential calculus. 

All of this work, which was of first-rate importance, was held 
up because of Huygens's attempts to construct a successful 
marine clock a task which was obstructed more by the limita- 
tions of the artisan's resources than by theoretical difficulties. 
Alexander Bruce, Earl of Kincardine, then living at the Hague, 
collaborated with Huygens in this work. In January 1663 Bruce 
crossed to England with two pendulum clocks suspended from 
ball and socket anchorages in the ceiling of his cabin. The 
weather was so rough that one clock fell from its suspension and 


the other stopped also. In April of this year two similar clocks 
were taken on a voyage to Lisbon by Captain Holmes. One clock 
went fairly regularly, and the report on its behaviour is preserved 
in the British Museum. A filibustering expedition in 1664 to 
the west coast of Africa and Guinea gave Holmes another 
opportunity. On one occasion the clocks proved more accurate 
than the method of dead reckoning in use. Huygens was 
optimistic as a result of this report and quoted it in his 
Horologium Oscillatorium of 1673. In 1665 war broke out 
between Holland and Britain and this ended English collabora- 

Huygens rather resembles Hooke in the variety of his 
scientific interests, but he was far more thorough than his 
English contemporary and was besides a more " mathematical 
head ". Besides his work on the marine clock and in mechanics, 
his work on telescopes and the theory of optics was kept up. It 
is worth noticing, this dual activity experimental and mathe- 
matical. Huygens used it to obtain a guiding idea rather than 
a quantitative result. Very great obstacles then lay in the way 
of exact quantitative work except in Astronomy. Huygens thus 
knew experimentally what order of aperture was needed, in a 
telescope of given length and magnifying power, to produce a 
clear and sufficiently bright image. He saw that the building of 
longer and yet longer telescopes required improved methods of 
lens grinding so as to secure sufficient aperture. It also raised 
problems of a purely structural kind. Wooden tubes, suitably 
braced, were used for telescopes of about ao or 30 feet; for great 
lengths Huygens proposed using two short tubes, one at the 
objective and one at the eyepiece and with rings placed along 
the intervening space. This method was tried in Paris, but it was 
found to be very difficult to align the two lenses as can be 
imagined. These " aerial telescopes " gave high magnification 
but poor definition. Many astronomers accordingly experi- 
mented on the grinding of lenses to forms suggested by 
Descartes. Many abortive attempts were made before the idea 
was abandoned. Huygens made some use of a machine for 
grinding lenses but it was not possible to make really large lenses 
by any method then known. His superior knowledge enabled 
him to see that the eyepiece could in certain ways be made 
to compensate for the defects of the objective. In 1662 he spoke 
of using two oculars instead of one as a " new manner " of en- 


larging the field of view. The date of his well-known eyepiece 
is, however, not quite certain, but it probably was not invented 
much before 1662 and it may have been as late at 1666. 

Huygens's theory of Saturn's phases was at this time so widely 
accepted, and was confirmed by observation as improved 
telescopes came more and more into use, that only more detailed 
matters remained to be settled. It was questioned, for example, if 
the periods of the phases agreed with the theory that the ring 
remained at a constant inclination to the ecliptic. Huygens 
showed how the phases could be calculated and succeeded in 
converting most of his critics. Wren wrote that " when . . . the 
Hypothesis of Huygens was sent over in writing, I confesse I was 
so fond of the neatness of it and the naturall simplicity of the 
contrivance, agreeing so well with the Physicall causes of the 
heavenly bodies that I loved the invention beyond my owne ..." 
And the accumulation of observations bore gradual witness to 
the success of Huygens's work. 

Wren's hypothesis is now forgotten. He and Neile, in 1658, 
tried to reproduce the appearance of Saturn by fitting an ellipti- 
cal " corona " to the planetary globe, meeting it at two places. 
They suggested that this corona rotated with the planet once 
during its revolution round the sun, on an axis coinciding with 
the plane of revolution. 

As a result of his work in astronomy and in mechanics 
Huygens's reputation was already high. Moreover, his scientific 
temper was in accord with that of the best spirits of his age. 
" I notice," he wrote to Boulliau (who was an ardent Pythagorean) 
on receiving a copy of his treatise on light " that in many places 
you dispute the opinions of Aristotle. That is always worth 
doing." His opposition to the Aristotelianism of the schools, his 
disregard of the Catholic opposition to Copernicanism, and the 
steadfastness of his belief in the new mathematical method 
brought him the esteem of modern spirits among his contempor- 
aries. It was, then, natural that in France, where Colbert was 
working to raise the achievements of art and learning above that 
of previous ages, Huygens should be considered as one of the 
more brilliant among notable foreigners who might be invited to 
take up a residence in Paris. 

Just as we know little about Newton which is not a descrip- 
tion of his mental quality, so it is with Huygens. His corre- 
spondence speaks a mind of great intellectual power and clarity. 


and the singular absence of violence and prejudice in his com- 
ments on men and things is but a necessary concomitant of this 
mentality. Nevertheless, he was a very human creature and one 
can sense that the parental authority at times aroused irritation 
just as at other times the elderly Constantin's desire to show off 
his son caused amusement. The trouble was that old Huygens's 
attitude to his sons did not change as they grew up into men, and 
when he was forced to treat them no longer as children he 
regarded them as young diplomats who might conveniently do 
him services in different parts of Europe. Diplomacy was, how- 
ever, not much in Christian's line, and while he affected French 
elegance and a seriousness of bearing, at the same time he was 
impatient with those who were tedious and self-important. 


The vicissitudes which both the French and English societies 
experienced before receiving official support, and even after, were 
such that they might well have died in infancy. Meetings at 
Montmor's were discontinued in 1661, but in 1662 the society 
held meetings at the house of the Marquis de Sourdis and when 
Huygens made another short visit to Paris in 1663 the society had 
regained much of its former activity. After the foundation by 
charter of the Royal Society in 1662 it was inevitable that in 
Paris, where meetings had been held at Mersenne's and else- 
where as early as 1650, the idea of a similar institution should be 
discussed. Sorbiere, ignoring or ignorant of the early history of 
the Royal Society, considered that the early Paris societies had 
in fact led the way. But however this may be, he and Huygens 
seem to have been on a semi-official errand when they came to 
London in 1663 to study the organization of the new Royal 
Society. Writing to Boyle of Huygens's introduction to the 
Royal Society, Oldenburg said " we had no ordinary meeting; 
there were no less than foure strangers, two French and two Dutch 
gentlemen: ye French were, Monsieur de Sorbiere and Monsieur 
Monconis; ye Dutch, both the Zulichems, 1 Father and Son, all 
foure inquisitive after you." Huygens evinced some surprise that 
no particular qualifications appeared to be necessary for election 
to the Royal Society at this time. Christian accompanied his 

* Christian Huygens held the title of seigneur de Zulichem (in the province 
of Gueldre) up to the death of his father. He then inherited the title of 
seigneur de Zeelhem. 


father on a diplomatic mission on this occasion and was still in 
London when the news of his award from Louis XIV was made 
public. This necessitated a return to Paris. 

During this short stay in England, however, he had occasion 
to see further into the character and customs of his hosts. 
Through his father's connections in this country he dined a good 
deal with the great, and met many personalities outside the 
scientific circle at Gresham College. Huygens and his brother 
Constantin both dabbled in art, and perhaps his chief interest on 
this occasion lay in his visits to the studio of Sir Peter Lely from 
whom he obtained a recipe for making pastels. 

Soon after his return James Gregory arrived from England 
with some correspondence from Moray. Moray wrote of this 
young man that he had " a present to make you of a book of 
which he is the author, which he calls Optica Promota . . . " He 
suggested that Huygens should give his opinion of the work and 
its author but Huygens left no record that he did this. The work 
was interesting in that it contained a description of a reflecting 
telescope some eight years before Newton's invention. 

After wintering in Paris, Huygens returned to the Hague 
(1664) bent on the pursuit of more fundamental researches than 
could be carried out in Paris. Not until 1666, when he became an 
official member of the newly formed Academic Royale des 
Sciences, did he return. Although, therefore, Huygens's work for 
the new academy was very important and the prestige he con- 
ferred on it was especially advantageous, it fell to others to com- 
plete the details of the organization. A wealthy amateur named 
Thevenot gave hospitality to the society at this time and did a 
good deal of the preliminary organization. It was he who, doubt- 
less with Colbert's knowledge, approached Huygens in 
November 1664 with a suggestion that he should become a 
member of the reconstituted society. As will be seen, the offer 
finally carried with it an official position in Paris with facilities 
for scientific work. 

Soon after his return to Holland, Huygens set about obtain- 
ing patents protecting his design of a pendulum clock for use at 
sea for determining longitudes. The news of this move not un- 
naturally aroused a good deal of excitement. The commercial 
value of a reliable method of finding longitudes at sea would be 
enormous and several others were after the prize. The members 
of the Royal Society, who knew of previous trials with marine 


clocks, were frankly sceptical about the use of a pendulum clock. 
As a result of full discussion, the society, with the national 
interest in view, resolved to investigate other methods. Of these 
some sort of spring-regulated clock appeared to be the most 
promising. It is not surprising, therefore, that Hooke, a most 
fertile experimenter, should have taken up the question of the 
isochronism of the oscillations of a loaded spring. In August 
1665 Huygens heard of Hooke's successful experiments and his 
confidence that a spring-regulated clock would be the solution of 
the problem. 

Huygens returned Hooke's scepticism. So long ago as 1660, he 
remarked, the Due de Roannes had tried the idea but without 
success. Temperature changes, he considered, would have a 
serious effect on the going of such a clock and sufficient accuracy 
would be impossible. Hooke, he concluded, spoke too confidently 
about this " as also of many other things ". Nevertheless, 
Huygens tried a spring-regulated clock in November 1665, but 
was hindered through the great delicacy of workmanship 
required. Brouncker, in England, found that Hooke's spring 
driven spring-regulated clock was not so accurate as a pendulum 
clock. The plague interrupted scientific work in London and 
Hooke's Potentia Restitutiva, on the properties of springs, did 
not appear until 1678. Huygens had to leave the Hague and 
retire into the country for a time. 

There, at Voorburg, he returned to his work on the com- 
pound pendulum, in particular the problem of determining the 
centre of oscillation. Lacking a general method, he proceeded 
to study the problem inductively, starting with several simple 
examples. It was not long before, discarding the erroneous work 
of Descartes, he arrived at some " quite pleasant propositions ". 
The work aroused great interest in England and its technical 
nature will be explained later. 

It should be mentioned that by this time (1665) there were two 
scientific journals of repute for the publication of new work. The 
publication of the Royal Society, Philosophical Transactions, was 
begun by the secretary, Oldenburg, on his own initiative in 
March 1665; in Paris the Journal des Savants was started, also as 
a private venture, by de Sallo in January of the same year. De 
Sallo's privilege was withdrawn after about a year because of his 
denunciation at Rome, but the Abb6 Gallois restarted the journal 
in January 1666. Neither the Journal nor the Transactions had 


the form of modern scientific periodicals: little original work 
was published and they were more of the nature of reports. The 
second number of the Transactions bore an account of some 
observations by Guiseppe Campani on Saturn's ring. These were 
of interest since Campani claimed to have distinguished the 
shadow thrown on the planet by its ring, the remarkable thing 
really being that his telescope was sufficiently good for such 
detail to be seen. It was said that Campani's lenses were ground 
and polished on a machine, but attempts so far made in this 
direction had been discouraging. Hooke published an account 
of a machine but it does not appear to have been well tested and 
he was castigated for publishing an account " upon a meer 

Cassini, using a telescope made by Campani, observed a per- 
manent mark upon the surface of Jupiter and from its return was 
able to give the period of revolution. This, Huygens affirmed, 
was " assuredly a very fine discovery ". He himself succeeded in 
observing the shadow of one of the satellites of Jupiter on the 
surface of the planet as predicted by Cassini. He also spent some 
time studying a comet which made its appearance at the end of 
1664. As will be explained later, the paths of the comets, so far 
as these were known, were proving a great difficulty for 
Descartes's cosmology. Huygens was primarily interested in them 
as they concerned the Copernican theory. He was at first sceptical 
about the idea that they recur at long intervals of time. It is 
interesting that Horrox's defence of the Copernican theory, 
written about 1635 and resuscitated by members of the Royal 
Society, came to Huygens's notice through his correspondence 
with Moray. Horrox, although he died at the age of twenty-two, 
is generally agreed to have made his mark as an astronomer of a 
very high order. 

The largest telescopes used at this time were of the type now 
known as Huygens's aerial telescopes, but it is not clear that he 
originated the idea or wished to claim it as his own. Auzout 
used an aerial telescope and devised his own method of aligning 
the lenses; in England the suggestion was widely attributed to 
Wren. An invention of greater importance and one to which 
Huygens made an interesting contribution was the micrometer 
eyepiece. It began to be realized that telescopes could be used 
for the determination of small quantities which were completely 
beyond the scope of ordinary instruments used up to that time 


Huygens's Clock as the Centre Feature of a design showing 
Scientific Apparatus of 1671 


for quantitative work. The measurement of small angular 
separations, for example, required the use of a very large 
quadrant, but these large instruments became distorted under 
their own weight. Gascoigne first hit on the idea of using two 
fine hairs close together and situated in the focal plane of the 
objective as a means of converting the telescope to quantitative 
measurements. Auzout and Huygens did some measurements 
of planetary diameters in 1664 an d 1665, but Huygens's micro- 
meter was a thin plate of metal in the form of a trapezium. It 
was inserted between the two lenses of his eyepiece where the real 
image was formed. The plate could be moved until the disc of 
the planet was just obscured. In this way, as early as December 
1659, h e obtained a good result for the diameter of Mars. Shortly 
after Huygens went to Paris in 1666 a micrometer consisting of 
moveable hairs was used. The modern form of micrometer was 
invented by Auzout and Picard. Curiously enough it was Picard 
who saw the value of the pendulum clock in astronomy rather 
than Huygens. Delambre remarks that Huygens " started the 
great revolution " in practical astronomy by his invention of the 
pendulum clock but it was Picard who did most to introduce 
regular time observations at the Paris observatory. Using 
Huygens's pendulum clock he used the times of meridian transit 
of stars to determine their differences in right ascension. 

Huygens was not, in fact, a regular observer. His contribu- 
tion to astronomy lay rather through his work on optics, which 
had throughout a practical bias: the invention of his eyepiece 
and the study of conditions under which spherical aberrations 
may be reduced. In this period just preceding his departure for 
Paris, Huygens became deeply interested in two works sent over 
from England : Hooke's Micrographia (1665) aad Boyle's Experi- 
ments and Considerations touching Colours (1664). Hooke, 
indeed, was at his best in descriptive and experimental work in 
which mathematics was not required. The hypotheses which he 
and Boyle advanced regarding the nature of light and the cause 
of colours were extremely stimulating and aroused Huygens to 
the desire to carry out experiments on the subject. From these 
days some of his important work in physical optics may be dated. 
He was convinced that before the phenomena of colour could be 
explained it would be essential to understand the mechanism of 
refraction. This, he considered, Hooke and Boyle had omitted to 
study sufficiently. His own note-books show that he calculated 


the order of thickness of the air film involved in the production 
of colours by interference in the so-called Newton's rings experi- 
ment (November 1665). Boyle, while admitting that he knew of 
this experiment, wisely declined to be drawn into a discussion of 
its explanation. 

In his views on the nature of light Huygens always showed a 
greater dependence on Descartes than in the rest of his work. 
This bias may explain his first scornful reception of Fermat's 
least-time principle, for Fermat was, of course, the great critic 
of Descartes 's work in optics. His principle that a ray of light 
follows that path for which the time of transmission is less than 
for any alternative path had also an Aristotelian flavour, or so it 
seemed to Huygens. He declared he found no satisfaction in the 
idea and considered it was a " pitiable axiom ". Nevertheless, he 
repeated Fermat's calculation of indices on this " obviously pre- 
carious " principle and, while retaining doubts as to its validity, 
began to be convinced more and more that the refractive index 
of a medium is in fact given by the ratio of the velocities of light 
in air and in the medium. It was necessary to suppose, with 
Fermat, that light has a finite velocity, whereas Descartes staked 
his scientific reputation, as he said, on the belief that its velocity 
is infinite. Roemer's famous calculations of 1676-7 were thus 
extremely important, for they showed that Fermat and Huygens 
were correct. 

In the meantime, as has been mentioned, the men of science 
in Paris had not found it easy to get the project of a permanent 
academy of science properly launched. The intimations Huygens 
received of a position in such an academy were not for a time 
followed by any concrete offer. Nevertheless, his name was kept 
in front of Colbert. Moray wrote to Oldenburg in 1665 that 
"Colbert intends to sett up a Society lyke ours and make 
Huygens Director of the designe," but during this year Huygens 
began to feel far from confident about the statements which 
reached him from Chapelain. He bombarded Carcavy with 
anxious letters and his feelings had to be assuaged with a variety 
of excuses. No doubt official delays occurred and accommoda- 
tion had to be found. Huygens was, however, more concerned 
over the amount of his salary, clearly through anxiety to live in 















the style to which his upbringing and habits had accustomed 
him. When he arrived in Paris in 1666, it was to find that no 
plans for the new society had been drawn up: The official found- 
ing of the Academic Royale des Sciences, on June ist, meant at 
first nothing by way of financial aid. It was simply agreed that 
Auzout, Roberval, CarcAvy, Frenicle, Picard, Buot and Huygens 
should be the nucleus. But, from a letter l written by Montmor 
to Alessandro Segni, the secretary of the Accademia del Crusca, 
it looks as if it was from the first intended that Huygens should 
have a prominent place. Then, too, writing to Hevelius in 1667, 
Boulliau said: "Chief of all is the renowned Christian 
Huygens . . . Next are Roberval . . . Auzout ..." 

Huygens became a close friend of the Colberts and served on 
occasion as the minister's scientific adviser. Meetings were held 
at first in Colbert's library and the first co-operative undertaking, 
an observation of a lunar eclipse, June 16, 1666, was planned to 
take place at his house. Unfortunately, cloudy weather made 
observations impossible. But two weeks later the same group, 
Huygens, Carcavy, Roberval, Auzout, Frenicle and Buot met to 
observe a solar eclipse. Once more visibility was poor and results 
were disappointing. Micrometer measurements giving the rela- 
tive diameters of the sun, moon and planets were, however, made. 

The appearance of the members at these early gatherings 
of the Academic Royale has been excellently recorded in the 
work of the skilful engraver Lfe Clerc. One of this artist's 
pictures, for example, shows an informal meeting of members 
and may well represent an hour spent in desultory discussion 
before the giving of an address. It has been suggested that the 
figure holding a lens and standing in the window is that of 
Huygens. When we look at the plate showing a meeting 
attended by Louis XIV (facing p. 60) it does not appear that 
Huygens was included. This is the opinion of E. C. Watson,* 
who points out that Huygens was away from Paris through 
illness early in 1671. 

In August 1666 Huygens took over apartments at the 
Bibliothique du Roi after the headquarters of the Academic had 
been transferred there. On December M the society gathered 
officially for the first time to hear from Carcavy the decision of 
the king to protect the new institution. 

i A. J. George. Annals of Science, III, 37*. 
* E. C. Watson, Osiris, VII, 556. 


Adherents of Descartes's philosophy, men, that is, who gave 
out their belief in vortices of subtle matter and who did 
not accept atoms or the existence of a vacuum, were not con- 
spicuous in the make-up of the Academic. Roberval was a noted 
critic of Descartes; later the Academic included Mariotte, who 
also was dubious about Cartesian theories. Frenicle resembled 
Mariotte in being prepared to accept resemblances between facts 
without feeling obliged to attribute them prematurely to a single 
cause. Huygens alone referred to the doctrines of Descartes's 
Principia when called on for explanations of such phenomena as 
gravity, and he and Charles Perrault for a time exerted a slight 
influence in favour of Descartes. In time, it is clear, Huygens 
became distinctly aware of the failure of such an experimen- 
talist as Rohault to maintain his Cartesian explanations without 
disingenuousness,and his work shows a progressive decline in his 
adherence to the great " system ". Yet he long remained in two 
minds and it only required some ignorant criticism of the great 
philosopher to rouse him to his defence. It is surprising too, to 
see how closely the form of some of his work (for example, that 
on impact) resembled that of Descartes 1 . 

Unfortunately this period of Huygens's work in Paris was 
twice interrupted by serious illness, necessitating a return to his 
native air. One gains the impression that his position was thereby 
weakened in some degree, for his absences were prolonged. When 
he left for yet a third time to regain his health it was never to 
return. His protector Colbert died soon after and profound 
changes in the p critical situation militated against his regaining 
a position which all along had aroused a certain envy. Huygens's 
last years were consequently spent in comparative retirement. 
Mach says that Huygens " shares with Galileo a noble, unsur- 
passable and complete uprightness " and this is a true estimate. 
The manner in which his years at Paris came to an end can only 
be deplored. Huygens's scientific work throughout illustrates a 
readiness to make his personal reputation always subservient to 
larger interests. Nil actum reputans, si quid superesset agendum 
was, according to the historian PoggendorfF, his adopted device. 

The presence of Huygens in Paris throughout the onslaughts 

of the French armies against the Dutch Republic is the fact 

which historians find hardest to explain. To a large extent wars 

reflect nothing of individual feeling towards members of another 

l Cf. Mouy. Le Dtveloppement de la Physique Cartdsienne 1646-1712 (p. 197). 


Louis XIV at a Meeting of the Academic 


nation and in those days the instruments of propaganda 
necessary for whipping up appropriate hatreds did not exist. 
Nevertheless, in the shifting scene of the wars of Holland, now 
against England with France as ally, now against France, then 
against both, and later with England as an ally, the opposition to 
Louis XIV really remained the one permanent feature. Louis 
was bent on destroying the Dutch Republic and, with the 
treacherous help of Charles II, it looked in 1672 as if he would 
succeed. Huygens could have secured a position of eminence 
under the Prince of Orange at this time but he had a deep 
repugnance for political activity and remained in Paris, suspected 
by some but protected throughout by the minister Colbert. It 
is not surprising, therefore, that he came in for some criticism by 
his fellow-countrymen. This criticism was brought to a head in 
1673 by the eulogistic dedication of his great Horologium 
Oscillatorium to Louis XIV. The explanation of these facts 
seems to be that, once having yielded to the cordial friendship of 
his associates at the Academic Royale and having decided to 
endure the war, he had to pursue a difficult and always depress- 
ing course. The dedication may be regarded as a piece of 
political wisdom, as justifying his continued patronage, in fact. 
As a friend of the Dutch ambassador van Beunigen, who during 
the short war of 1667-8 was suspected of a plot against Louis, it 
would have been easy for him to come under suspicion as a spy. 
For Huygens was by no means remote from the world of affairs. 
He was very well known at court and had many influential 
friends. After the rise of Louvois and the death of Colbert none 
of these things mattered; the feelings of Louis towards the house 
of Orange can scarcely be said to have improved after 1678. 

But in 1666 Huygens was indisputably the one who chiefly 
guided the affairs of the Academic Royale des Sciences. Profiting 
from his knowledge of the Royal Society, Huygens emphasized 
in Paris the importance of Bacon's teaching. " Experiment and 
observation/' he wrote, " provide the only way of arriving at the 
knowledge of the causes of all that one sees in Nature." This 
attitude is the more striking when one reflects that Descartes 
had so long been his model. The point here is that while both 
Bacon and Descartes distrusted formal logic, Descartes scorned 
empiricism while Bacon apprehended its power. It is not clear 
that Huygens realized the shortcomings of Bacon's " method ". 
The great omission in Bacon's scheme of research was the recog- 


nition that measurements provide the key to the understanding 
of phenomena. Bacon ranged himself with Aristotle in saying 
classify when he should have said measure. 

The attention paid to Chemistry should, in Huygens's view, be 
restricted to essential problems. He appears to have recognized 
that the old alchemy was decadent and that the beginnings of 
a true science lay in the work of Boyle and others. Problems 
such as that of combustion were obviously worth the closest 
study. Huygens was interested in Hooke's experiments at the 
Royal Society which, he held, agreed with the " bizarre hypo- 
thesis " of an " aerial saltpetre ". This hypothesis of an active 
constituent in the air he considered was " not ill conceived " but 
he tended towards Moray's empiricism. " We others/' wrote the 
latter, " look for the truth of existence and the nature of things 
as belongs to the true philosophy ". 

There can be no doubt of the influence of the London group 
on Huygens 's views of the functions of the new Academy. This 
influence abroad was recognized by the English themselves, who 
were fully conscious of the unique importance of their work. 
" I hope our Society will in time ferment all Europe at least," 
wrote Oldenburg to Boyle. " Let envy snarl," he wrote, when 
the new societies excited opposition, " it cannot stop the wheels 
of active philosophy in no part of the known world." 

The Academic at first made astronomy its special study, en- 
couraged, no doubt, by the occurrence of a partial eclipse of the 
sun in 1666. Huygens noted with dissatisfaction the paucity of 
astronomical observations in earlier years and this was to be 
remedied. New observatories were in the course of construction 
at Greenwich and Paris; Hevelius at Danzig had for some years 
applied himself to completing Tycho Brahe's observations and 
had, in 1661, made with Boulliau careful observations of a solar 
eclipse. In 1666 more was expected from eclipse observations, 
namely, to rectify the motion of the earth and the moon and to 
determine differences of meridian on the earth. In the course of 
the work, several telescopes were compared and micrometers were 
used for obtaining the relative diameters of the moon and sun. 

In continuing his work on lenses at Paris, Huygens was hin- 
dered by the poor quality of the French glass, which was inferior 
to the Venetian. The material showed veins or striae and tended 
to extrude salts on cooling. Lens-making brought Huygens into 
contact with the work of Spinoza, who had then a greater reputa- 


tion as a lens-grinder than as a philosopher. Lenses were ground 
by hand in a hollow form or mould in which abrasives of increas- 
ing fineness were successively used. It was the impossibility of 
grinding any but small lenses in this way that put a limit to the 
power of telescopes. The appearance of colours in the image was 
considered by Huygens to be connected with the inclination of 
the lens surfaces. The error in this was recognized by Newton 
some years later but in the meantime a great deal of work was 
expended in the attempt to make lenses of other than spherical 

Astronomy in Paris gained very greatly by the arrival of 
Cassini in 1669. His first observations were made at the new 
observatory in 1671. Here he continued his striking work on the 
rotation of certain of the planets. Huygens had observed the 
rotation of Mars in 1659 ^ ut true to his device, had not con- 
sidered the results sufficiently good for publication. Cassini was 
rewarded by the discovery of four satellites of Saturn and the 
division in Saturn's ring which is now known by his name. 

Huygens was not a competitor with Cassini for the honours of 
new astronomical discoveries. After 1666 his interests lay more 
in the direction of terrestrial mechanics and, as the sequel shows, 
this preference was sound. " I am now starting experiments on 
circular motion," he told his brother in 1667. A few years later, 
when Richer's expedition to Cayenne returned to Paris it brought 
back interesting evidence which bore on the question of the 
earth's gravity, but the effect of circular motion of a medium 
was the question which at this time interested Huygens, for it 
was through this that he hoped for an explanation. In taking up 
the effects of rotation, Huygens was, from one standpoint, return- 
ing to work which he had put away ten years before. By 1659, 

v 2 
it is thought, he had arrived at the expression for the 

acceleration towards the centre in the case of a body describing a 
circular path. In 1669, however, he chose to address the Academic 
Royale not on this but on an elaborate theory employing the 
vortex of subtle matter as the cause of gravity. Any easy con- 
victions we may have that Huygens had by this time rid 
himself of Cartesian influences must be profoundly shaken by 
a perusal of this discussion. The opposition with which his 
theory was greeted by Mariotte and Roberval on this occasion 
may have been highly beneficial, for the criticisms they made 


were entirely justified. It may be mentioned in passing that 
Huygens at this time believed that circular motion is a funda- 
mental form. Uniform rectilinear motion, he saw, had no effects 
on events which normally occur in an apparently stationary 
environment. Circular motion, howfever, introduced new effects. 
It was only after the appearance of Newton's Principia that 
Huygens retracted this statement of the absolute nature of 
circular motion. He then, more consistently, took a firm stand on 
the relative nature of all motion and against the idea of any 
absolute space. 

Mariotte was a French priest who joined the Academic 
Royale in the year of its foundation and thereafter played an 
important part. He must be reckoned among the lesser lights 
who at this time were attempting to make the important next 
step beyond the mechanics of Galileo. His TraiU de la percussion 
ou choc des corps (1677) shows that he and Huygens were work- 
ing on similar problems. When Oldenburg approached Huygens 
in 1668 with a request that he should contribute to the Royal 
Society some work on mechanics he replied by sending some 
work on impact. This, afterwards published in the posthumous 
Tractatus de Motu Corporum ex Percussions (1703), is really a 
study of various applications of the law of conservation of 
momentum. There can be no doubt that Newton profited from 
the work on impact which was carried out by Huygens, Mariotte, 
Wallis and Wren. The formulation of his relation between rate 
of change of momentum and external impressed force completed 
in a magnificent way this contemporary work. In regard to centri- 
fugal force Huygens forestalled Newton by many years. " What 
Mr. Huygens has published since about centrifbgal force I sup 
pose he had before me," wrote Newton with some chagrin. 

The immediate result of the correspondence with Oldenburg 
was that Huygens learnt that Wren and Wallis had both com- 
municated papers on the subject of impact and momentum and 
at practically the same time as his own. More instances were to 
come in which Huygens felt himself to have been unfairly fore- 
stalled in publication and in some cases he gave vent to severe 
criticisms which were by no means justified. Oldenburg showed 
great fairness and removed much oif this feeling of resentment, 
but the outbreak of some acrimonious correspondence over some 
mathematical work by James Gregory shows that Huygens had 
become unduly nervous for his reputation. When Mercator put 


forward a method of determining longitude by means of a pen- 
dulum clock he roundly condemned his intrusion. It was for- 
tunate that when Barrow's Lectiones Opticae came out in 1669 it 
was evident that the work did not overlap with Huygens's pro- 
longed researches in optics. Huygens was surprisingly slow to 
learn the consequences of his own attitude towards publication. 

It was not customary in those days to isolate a particular 
problem and to study it exclusively for a considerable time. The 
seventeenth century men of science were nearly all capable of 
turning their attention to a wide range of subjects and they fre- 
quently were engaged on a variety of topics. Huygens indeed 
must be considered one of the most versatile men of the age, 
for he excelled Hooke in the quantitive nature of his work while 
at the same time he showed as wide a range of activity. Hooke's 
Micfogrophia stimulated Huygens at this time to attack the 
problems of constructing microscopes, employing the theoretical 
advances he had achieved in his work on the telescope. Spinoza 
was interested in similar problems. Galileo was described as 
having constructed " an occhiale which magnifies ... so that one 
sees a fly as large as a hen ". This was a compound microscope. 
Hooke improved the instrument as regards its mounting and the 
illumination used. Optical improvements were seriously needed. 
Huygens's own microscopic observations will be mentioned later; 
they belong to the years after his translation of Leeuwenhoek's 
work into French in 1677 or 1678. 

As has been mentioned, Huygens's health was never robust. 
From early youth he was from time to time subject to a certain 
kind of debility, later accompanied by severe headaches. The 
illness of 1670 brought about his complete prostration in Paris 
and he clearly believed himself to be at the point of death. In 
these circumstances he concluded that he should bequeath his 
more important unpublished work in mechanics to someone 
capable of appreciating its importance and he decided to send it 
to London in the hands of Francis Vernon, secretary to the Eng- 
lish ambassador. This action is sufficiently interesting in view of 
Huygens's official position in Paris for Vernon's account to be 
given at length. In a letter to Oldenburg he described Huygens's 
condition "... I saw the condition hee was in which was none of 
the most lively, that his weaknesse & palenesse did sufficiently 
declare how great a destruction his sicknesse had wrought in his 
health and vigour & that though all was bad, which I saw, yet 


there was something worse which the eye could not perceive nor 
sense discover, which was a great dejection in his vital spirits, an 
incredible want of sleep, which neither hee, nor those who coun- 
celd and assisted him in his sicknesse knew how to remedie & 
that hee did not know what the end of these things would bee, 
butt his fancy was ready to suggest the worst ..." This mood 
Vernon did his best to dispel. He accepted his mission to England 
should the worst befall. Then, he wrote, " hee fell into a discourse 
concerning the Royal Society in England wich hee said was an 
assembly of the Choicest Witts in Christendome & of the finest 
Parts: hee said hee chose rather to depositt those little labours 
of his which God had blest and those pledges which to him were 
dearest of anything in this world, in their hands sooner than in 
any else. Sooner then of those into whose Society hee was here 
incorporated & from whom hee had received all demonstrations 
of a most affectionate civilitie because hee judged the Seat of 
Science to bee fixed there & that the members of it did embrace 
& promote Philosophy not for interest, not through ambition or 
a vanity of excelling others not through fancy or a variable 
curiosity, butt out of naturall principles of generosity, inclina- 
tion to Learning & a sincere Respect and love for the truth. . . . 
Whereas hee said hee did foresee the dissolution .of this academic 
because it was mixt with tinctures of Envy because it was sup- 
ported upon suppositions of proffitt because it wholly depended 
upon the Humour of a Prince & the favour of a minister, either 
of wich coming toe relent in their Passions the whole frame & 
Project of their assembly cometh to Perdition/' 

It is clear that so early as 1670 differences had arisen between 
Huygens and certain members of the Paris Academy. This fact 
will be of interest later when the circumstances of the rupture 
of his official connection are considered. 

Huygens's illness lasted in acute form for several weeks during 
which great anxiety was felt by his friends in Paris and London. 
In June there were signs of recovery and three months later 
the convalescent was able to return to the Hague. In October he 
resumed correspondence with Oldenburg. 


Apart from this winter in Holland, 1670-71, the five years 
from 1670 to 1675 were spent by Huygens in Paris. And they were 
stirring years in the scientific world. Huygens as chief of the 


Paris Academic was at the centre of things on the Continent, 
while he was well informed of what passed in England. In 167 1 
Picard's M&ure de la Terre came out, a work of interest from 
the technical aspect as well as for a general discussion of current 
theories concerning the shape of the earth. Picard, for example, 
was well aware in 1671 that the length of a seconds pendulum 
was different at London, Lyons and Boulogne, but while he 
admitted that the results might be in conformity with the rota- 
tion of the earth, he did not think there was sufficient confirma- 
tion of the results, as yet, to justify any conclusion. Very probably 
he was acquainted with the notion of centrifugal force through 
his relationship with Huygens, for the latter had arrived at his 
important theorems as early as 1659. I* 1 l fy* news came from 
England of Newton's work on the solar spectrum; from Holland 
in the same year came interesting mathematical work by Slusius 
on the drawing of tangents to curves. 1673 was the year of Huy- 
gens's Horologium Oscillatorium, his magnum opus. In 1674 
Hooke issued a work giving his views on evidence for the 
motion of the earth. 

During this period Huygens worked with Denis Papin on 
the use of gunpowder as a source of useful energy, and, more 
important, with Leibnitz at Mathematics. In 1675 Leibnitz 
brought out his calculus differentialis. And to these busy years 
belong also the invention of the spiral-spring regulator and 
balance wheel which are essential parts of the watch and chrono- 

Yet they were not altogether happy years for Huygens. After 
the invasion of the Low Countries, by the armies of Louis XIV 
in 1672, he may frequently have asked himself why he had re- 
turned to Paris in 1671, and immersed himself in work as the 
only outlet for his despair at his situation. Huygens felt keenly 
the wrong done to his country and it required much tact and 
consideration from his friends in Paris to preserve the calm rela- 
tionship in which he had been accustomed to live. Huygens 
followed the course of the war with anxiety, but it must be 
remembered that at forty-three he was by upbringing and experi- 
ence almost as much a citizen of the French capital as he was of 
Holland. Paris was indeed the centre of the cultivated world and 
the prospects for the man of science who should be cut off from 
the activity of one or other of the two flourishing societies would 
be poor indeed. 


Huygens seems to have worked even when quite ill; certainly 
he was struggling to regain his old activity in October 1670 when 
he received some interesting work on mechanics from Wallis. 
Yet he agreed that his recovery (" for which I thank God ") was 
too recent for him to do other than attempt very little. 

We shall see that Huygens's work in theoretical optics, for 
example, rivals that in mechanics, but in 1670 he considered the 
latter subject more important. Though overlooked nowadays, 
the ideas he contributed to mechanics were as fundamental as 
his contributions to physical optics. His Horologium Oscilla- 
torium was practically completed and had grown from a treatise 
on the construction and regulation of the clock to a work on 
centres of oscillation, the tautochrone, the theory of evolutes 
and centrifugal force. As an examination of this treatise will 
show, it contains a great deal which was made more explicit in 
Newton's Principia (1687) although discovered by Huygens 
independently. The law known as Newton's First Law was known 
and used by Galileo and Huygens; Huygens, in addition, must 
have employed the Second Law in arriving at his propositions 
on centrifugal force as early as 1659. He also saw the necessity 
of distinguishing between mass and weight at about the same 
time (see p. 119). The greater merit of Newton's work, in 
fact, was that he gave a clearer presentation of these ideas and 
made them more useful by means of simple mathematical 

When the Horologium Oscillatorium came out in 1673, after 
Huygens's return to Paris, it showed the extent to which his 
thought had developed. The work was singularly free from Car- 
tesian influences. Huygens himself hoped that it would be in 
direct line with the great work of Galileo and his hopes were not 
disappointed. Newton wrote to Oldenburg of his " great satis- 
faction " with the work and said he found it " full of very subtile 
and usefull speculations very worthy of ye Author ". Newton 
especially admired Huygens's mathematical style and con- 
sidered him the " most elegant writer of modern times ". This 
remark starts some interesting reflections. Newton regretted that 
he had not applied himself to geometry before proceeding to 
algebraic analysis. It was Huygens's predominantly geometrical 
methods, employed in the Horologium Oscillatorium, which 
aroused his admiration. At this time Newton was well advanced 
in his work on fluxions and, as w-e know, Leibnitz took up similar 


problems after 1672. The central idea of the differential calculus 
owes a great deal to the study of motion, for this study intro- 
duced the notion of a continuously varying quantity. Huygens 's 
work in this connection was of the greatest importance for, as 
Leibnitz admitted, it was Huygens who had dispelled the rnys 
tery attaching to the study of motion. 

The two mathematicians, Huygens and Leibnitz, met in 
Paris in 1672 and Leibnitz became a regular visitor at the 
Bibliotheque du Roi. Under Huygens's guidance Leibnitz's ideas 
developed rapidly, for up to this date, as he himself admitted, he 
had been only an amateur in such studies. In 1674 Huygens was 
able to present to the Academic Royale Leibnitz's first paper on 
the differential calculus. Whether Huygens gave Leibnitz an 
inkling of Newton's work on fluxions will always remain an in- 
teresting speculation. Newton's own ideas date from about 1665 
or 1666 and there is no doubt that after 1669 these were well 
known to his friends in England. Wallis especially must have 
known about them. Huygens himself was not happy in the use 
of analytical methods. He was, in Newton's words, " the most 
just imitator of the ancients " and it is a striking fact that the 
classical geometrical method was used by Newton himself in 
writing the Principia. This fact, which has always troubled 
historians in some degree, must be explained by the prestige 
of Huygens at this time and the fact that proofs by the newer 
methods were not everywhere accepted. The ideas of both the 
Principia and the Horolofnum Oscillatorium were later cast in 
analytical form by the mathematicians of the eighteenth century. 

The Horologium Oscillatorium made a great impression on 
contemporary men of science. The propositions on centrifugal 
force, given at the end, were of course important in the develop- 
ment of planetary theory, and the conical pendulum interested 
those who, like Hooke, concerned themselves with the problem 
of time measurement. It is certain that Huygens employed the 
conical pendulum in clocks in 1659 and again in 1667, when he 
had more fully investigated the laws of motion involved. Con- 
troverting Hooke's claims to the invention of such a clock, he 
pointed out, what Hooke certainly did not know, that the conical 
pendulum should be so designed that all revolutions of the bob 
describe horizontal circles in the surface of a paraboloid of revolu- 
tion with the axis vertical. Only then would all revolutions be 


Huygens showed considerable dislike for quarrels such as the 
one of 1674 in which he became involved with Hooke and others. 
In 1675 he had yet another such experience. This was over his 
invention of the first successful spring regulator for the clock. 
Huygens's design of a spiral spring combined with a balance 
wheel is the one which is still used in watches and its distinction 
lies in the fact that the centre of gravity of the oscillating part 
bears a fixed relation to the stationary parts. This meant that the 
influence of gravity was eliminated. A quarrel with the clock- 
maker Thuret and the Abbe de Hautefeuille was settled by the 
Academic Royale on these grounds in Huygens's favour, for 
Hautefeuille used a straight spring and not a spiral. Hooke was, 
however, a more tedious antagonist in connection with the same 
invention and he was made more bitter by the support given by 
some members of the Royal Society for Huygens's priority. 
"Zulichem's spring not worth a farthing," he wrote in his 
diary. When Huygens, for a quiet life, gave Oldenburg the rights 
to the English patent, he drew from Hooke a full and free ex- 
pression of his long dislike of the secretary. Oldenburg, he 
averred, was Huygens's spy. " Saw the Lying Dog Oldenburg's 
Transactions," he noted eight months later, " Resolved to quit all 
employment and to seek my health." Oldenburg, he said angrily, 
was a "trafficker in intelligence". He would hear nothing favour- 
able about Huygens's marine clocks. " Seamen knew their way 
already to any Port," he somewhat childishly stated. Altogether 
the complaints of Hooke appeared in print over a considerable 
period tut they hold little interest now. 

A more famous result of Huygens's correspondence with the 
English scientists during his years in Paris was that the French 
became aware in 1672 for the first time of the work of Newton. 
On January 1 1 of that year Oldenburg wrote to Huygens of the 
" invention of a new sort of telescope by Monsieur Isaac Newton, 
Professor of Mathematics at Cambridge ". His next letter gave a 
full description and this was utilized by Huygens for an account 
published in the Journal des Savants of the following month. It 
should be mentioned that Gregory's design for a reflecting tele- 
scope was never put into practice and the new invention was 
based on his ideas. Huygens found it *' beautiful and ingenious " 
and he thanked Oldenburg for the news of " the marvellous tele- 
scope of Monsieur Newton ". The problem of making the con- 
cave mirror, though difficult, did not appear to be insuperable. 


He supposed that Newton had come to despair, as he had him- 
self, of overcoming spherical aberration but did not refer to the 
fact that it was this consideration which had led Gregory to his 
idea nine years before. Up to this date Huygens had not heard 
of Newton's experiments on the composition of white light and 
certainly underestimated the defect of chromaticism. 

It is consequently surprising, after this initial enthusiasm, to 
find that Huygens soon abandoned the reflecting telescope. The 
trials which he himself immediately carried out proved dis- 
appointing owing to the imperfect polish given to the mirror. 
Newton's first telescope was hardly more than a model and when 
the construction of larger telescopes was attempted in England, 
the most expert glass worker in London, Cox, found the task of 
grinding the mirrors too difficult. Reflecting telescopes of a large 
and useful size were in fact not made for more than fifty years. 
Huygens found that metal jnirrors were unsuitable since the 
polish was unequal to that given to glass and it was not perman- 
ent. He found himself compelled to return to the refracting tele- 
scope but realizing that a new difficulty beside that of spherical 
aberration now required to be overcome. 

For in March 1672 Oldenburg sent Huygens a copy of the 
Philosophical Transactions in which, he said, Huygens would 
find " a new theory of Monsieur Newton (the inventor of the 
cata-dioptric telescope) touching light and colours : where he 
maintains that light is not uniform but a mixture of rays of 
different refrangibility, as you will see fully in the same dis- 
course. . . " This copy of the Transactions contained, of course, 
an account of Newton's famous experiment on the spectrum. 
The Royal Society, on receipt of this, Newton's first published 
scientific paper, sent it to Huygens as the one whose opinions 
would carry the most weight. Huygens's reply was that the con- 
clusions drawn and the theory put forward seemed "very 
ingenious ". But, he went on, " it must be seen if it is compatible 
with all the experiments ". Three months later he wrote that he 
considered the compound nature of white light had been proved 
by Newton's experimcntum crucis, in which it was shown that 
the separate coloured rays emerging from the prism could not 
be further decompounded. Huygens went on, however, to make 
observations which disappoint the modern reader almost as much 
as they disappointed the young Newton. He questioned if it 
would not suffice to base an explanation " on the nature of move- 


ment " for the two colours yellow and blue only. Until the 
essential difference of these two colours was understood " he 
[Newton] will not have taught us what the nature and difference 
of colours consists of, but only this accident (which assuredly is 
very considerable) of their different refrangibility ". Failing to see 
the distinction between an impression of colour and the different 
rays of the spectrum, Huygens suggested that Newton would 
find that yellow and blue would be sufficient to produce white 
light. The other colours he regarded as " degrees of yellow and 
blue more or less deep ". 

These criticisms were an easy prey to Newton who, far from 
rushing prematurely into publication, had kept the work by him 
for at least five years. Oldenburg warned Huygens that Newton, 
then thirty, was not a man who spoke lightly about anything 
he advanced. Newton flatly denied that all colours could be " de- 
rived out of the Yellow and Blew . . . none of all those colours 
which I defined to be Original " could be so obtained. " Nor is 
it easier,." he insisted, " to frame an Hypothesis by assuming only 
two Original colours rather than an indefinit variety; unless it 
be easier to suppose, that there are but two figures, sizes and de- 
grees of velocity or force of the ^Ethereal corpuscles or pulses, 
rather than indefinit variety; which certainly would be a harsh 
supposition." It would be indeed, he remarked, " a very puzzling 
phenomenon ", "... But to examine how Colors may be ex- 
plained hy pot helically is besides my purpose. I never intended to 
shew, wherein consists the Nature and Difference of colors, but 
only to shew, that de facto they are Original and Immutable 
qualities of the Rays which exhibit them; and to leave it to 
others to explicate by Mechanical Hypotheses the Nature and 
Difference of those qualities: which I take to be no difficult 
matter ". It was unimportant if two colours in the spectrum 
could be combined to give an appearance of white. Such light 
was different in its physical nature from ordinary white light 
and could not be resolved by the prism into more than the 
two components. Clearly Newton was deeply disappointed, at 
the outset of his career, to receive so little appreciation of the 
true nature of this work from one so eminent as Huygens. He 
wished, he said, in future " to be no further solicitous about 
matters of Philosophy ". His rather summary answers to Huy- 
gens's remarks disposed at the same time of certain criticisms 
put forward by Hooke. For once Huygens and Hooke were in 


alliance, both opposed to what was essentially a new attitude 
to scientific problems; both, moreover, found the new 
facts difficult to reconcile with their respective wave or pulse 
theories of light. This first occasion for the comparison of Huy- 
gens and Newton raises, it will be seen, a question on which these 
two men of science consistently differed. This was the place of 
hypothesis in scientific method, a subject for which a later chap- 
ter must be reserved. 

It is unfortunate that the two greatest scientists of this period 
did not achieve harmony in their attitude to problems of com- 
mon interest. For their divergence was not particularly fruitful 
although it extended from optics into the realm of mechanics. 
Huygens greatly admired the Principia after its appearance in 
1687 and he met Newton in 1689. By r his time, however, the dif- 
ference of outlook had become too ingrained and Huygens at 
sixty had become less amenable to new persuasions. And yet 
this difference is certainly not so great as some writers have sug- 
gested. One biographer of Newton, Louis Trenchard More, con- 
siders that men like Hooke and Huygens relied on an inward 
sentiment of knowledge and in opposing Newton " were merely 
opposing theory by hypothesis ". Whatever the significance of 
this distinction, this is a question which can be dealt with only 
after a careful survey of Huygens's work as a whole. It will be 
seen that it is a profound mistake to treat Huygens as merely 
another Cartesian, for all his life he vacillated between the Car- 
tesian view that the objects of scientific calculation are products 
of thought and the materialism which regards them as external 
realities. Not only is there much to say in Huygens's favour with 
regard to the status of scientific concepts, but in methodology 
also Huygens perceived as well as Newton the end of scientific 
investigation. " I do not believe we know anything with com- 
plete certainty," he wrote to Perrault, " but everything probably 
and to different degrees of probability ... as 100,000 to i as in 
geometrical demonstrations/* The latter he considered were in 
a category by themselves. " In the matter of Physics there are 
no certain demonstrations and one can only know causes through 
the effects in making suppositions founded on experiments or 
known phenomena and trying afterwards if other effects agree 
with these same suppositions." These remarks should make clear 
the difference between Huygens and Descartes. For Descartes 
the intuitional method, to which More refers, did undoubtedly 


take precedence over the experimental. For Huygens, a life-long 
experimentalist, there was no way to final certainty. 

In the period we are considering Huygens continued his 
studies in optics but it is difficult to estimate his progress. This 
work began as early as 1652. By 1653 he had completed a first 
Tractatus de refractione et telescopiis of 108 pages. This was sub- 
sequently amplified but remained unpublished. As Huygens 
found, Cavalieri had independently obtained some of his results 
and published them in the Exercitationes Geometrical Sex. The 
only other previous writers of originality were Alhazen (nth 
century) and Kepler, whose Paralipomena was published in 1604 
and Dioptrice in 161 i . In England the most important work was 
done by Barrow and by Halley. The latter drew attention to the 
advantages of algebraic formulae; up to this time the relations 
used in optics were expressed in the most cumbersome geometri- 
cal form. Many of the problems to which Alhazen had given 
prominence were definitely geometrical in character and these 
continued to be studied. A new interest was injected into these 
somewhat academic studies by Bartholinus's discovery of double 
refraction in i66g. This was described in a short Latin treatise, 
Experiinenta Cryslalli Islandici Disdiaclastici . . . , which was 
widely read. Huygens obtained a specimen of Iceland Spar and 
a considerable part of his Traite de la Lumiere, completed in 
1678 (published 1690), deals with experiments he then carried out. 
Huygens had developed a pulse theory of light and the effort 
he made to reconcile his theory with the new and peculiar optical 
properties of Iceland Spar was a masterly one. Curiously enough 
there is little reference to this work in his correspondence. 

About this time Huygens had as his assistant Denis Papin, a 
Frenchman who later worked with Boyle in England. With 
Papin, Huygens in 1673 experimented on gunpowder as a source 
of mechanical energy. There is a possibility that Huygens had 
considered some kind of atmospheric engine as early as 1660 
when he talked with Pascal about " the force of water rarefied 
in cannons ". In thfcse experiments of 1673 we can see the fore- 
runner of Papin's atmospheric engine, which did in fact employ 
steam in place of gunpowder. Papin was later, through the in- 
terest of the Landgrave of Hesse, appointed professor at the 
university of Marbourg and it was here that he developed the 
atmospheric engine which gave Newcomen his clue. 

In exchange for Papin, as one might say, Oldenburg sent over 


to Huygcns the wealthy young amateur Walter von Tschirn- 
haus, a friend of Spinoza and Leibnitz. He belonged to a class 
which had early supported the new scientific societies but he was 
exceptional in that his interest in science did not quickly flag and 
in that he made himself something more than a mere dilettante. 
Through Tschirnhaus Huygens undoubtedly learned more of 
Spinoza's philosophical ideas but he showed himself little inter- 
ested in them. Unlike many of the seventeenth century men of 
science Huygens did not occupy himself with philosophical or 
theological questions and neither he nor Leibnitz seems fully to 
have grasped the nature of Spinoza's thought. 


Early in 1676 Huygens was again ill. There is no doubt that 
the illness of 1670 had recurred and this time he showed greater 
caution in meeting the danger. In March 1676 he set out to 
return home to the Hague while he was yet able, but the journey 
was slow and very uncomfortable. To his brother he confessed his 
doubts whether he would return to a life in Paris which seemed 
to be injurious, and even when he had recovered, a year later, 
he procrastinated under the pretext of uncertain health. Colbert 
gave permission for his remaining at the Hague for the winter of 
1677-78 and the return journey to Paris took place in June 1678. 

During these two years at home he pushed on with his re- 
searches. To these years belongs a great deal of work on the 
double refraction of Iceland Spar and the development of his 
wave theory of light. On November 122, 1676, Roemer read a 
paper to the Academic Roy ale in which he gave the first calcula- 
tion of the velocity of light. Huygens was immediately 
interested on receiving a copy of the paper and an interesting cor- 
respondence with Roemer was begun. The assumption that light 
travelled with a finite speed was fundamental in Huygens's work 
and on this assumption, he wrote to Colbert, he had " demon- 
strated the properties of refraction and, a little while since, that 
of Iceland crystal which is no small marvel of nature nor one 
which it is easy to understand ". It was consequently gratifying 
that this assumption should receive confirmation and that the 
speed of light should be approximately known. There was some 
similarity of outlook between Roemer and Huygens for Roemer 
supposed, like Huygens, that the passage of light through cry* 


tals (thrown into prominence by the Iceland Spar phenomena) 
was analogous to the transmission of impulses through con- 
tiguous spheres. The explanation of double refraction along these 
lines would, he considered, establish the truth of the theory of 
light which for some years had been associated with Huy gens'? 

It is well known that Huygens was led to his theory of trans- 
mission of light through his work on impact. The transmission 
of longitudinal, compressional, vibrations through perfectly 
elastic spheres seemed to him to have applications in light, since 
crystals and other transparent media might be supposed to be 
composed of assemblages of atoms. And even though he was 
unable to suppose the atoms of the elements were the actual 
medium since all substances are not transparent he found a 
mechanism which reduced light to a form of motion and brought 
it within the treatment of the " true Philosophy ". In this philo- 
sophy " one conceives the causes of all material effects in terms 
of mechanical motions. This, in my opinion, we must necessar- 
ily do, or else renounce all hopes of ever comprehending any- 
thing in Physics/ 1 This quotation is taken from the beginning 
of Huygens's Traite de la Lumiere. To explain the transmission 
of light through the Torricellian space and all manner of trans- 
parent substances, however, some pervading subtle medium was 
assumed. We must remember that Huygens was led to conclude 
that such a medium existed from his vacuum experiments. The 
ether was, however, not a continuous medium but was composed 
of very light particles in contact. These, on impact with the 
heavy vibrating atoms of incandescent bodies, transmitted their 
vibrations in all directions according to the laws of impact. The 
elasticity of air, Huygens thought, " seems to show " that it 
is made up of particles which are " agitated very rapidly in the 
ethereal matter composed of much smaller parts ". It was im- 
portant that slight impulses travelled as fast as strong ones, a 
fact which was readily explained by applying Hooke's law 
of elasticity to the particles of ether. Also, individual wave- 
lets by themselves were too weak to produce effects of light, 
which only arose when the wavelets combined to form a wave- 
front according to the well-known Huygens construction now 
given in all text-books on light. 

Huygens's theory is better described as a pulse theory rather 
than a wave theory but in the Traite he made the remark that 


the vibratory motion "... is successive and . . . spreads as sound 
does, by spherical surfaces and waves ", Hooke developed a wave 
theory also largely in relation to his observations of colours 
produced by thin films. The difference between his ideas 
and those of Huygens was mainly that Hooke did not consider 
the formation of a wave-front by the innumerable individual 

Not all scientists, however, were prepared to accept Roemer's 
estimate of the velocity of light. Descartes had been so con- 
vinced of the instantaneous transmission of light that he un- 
wisely said he would stake all his system of philosophy on its 
truth. Unlike Huygens there were many who remained under 
his spell. In the Traite, therefore, Huygens went to some pains 
to demonstrate the error of Descartes's reasoning. Cassini was 
opposed to Roemer's explanation of the apparent advance and 
retardation of the occultations of Jupiter's satellite, but mainly 
because only the innermost satellite had been studied. When the 
Academic had to decide on the dispute which arose over the 
work they came to the conclusion that Roemer was right; as he 
explained, the occultations of the outer satellites were less fre- 
quent and less sharply observable for obvious reasons. While the 
method he had put forward was the best one available for finding 
the velocity of light, he hoped that surface marks on Jupiter 
would prove of use, and later in the year observation of a spot 
on the planet gave the period of revolution of the planet on its 
axis. Observation of this spot could then be relied on in place of 
occultations for measurement of the velocity of light. Clearly 
Roemer was a man of the same outlook and ingenuity as Huy- 
gens. At the time when Huygens was at the Hague, Roemer was 
attempting to determine what effect the motion of the earth 
should have on the apparent positions of the heavenly bodies 
when this motion was transverse to the direction of the light 
rays. It scarcely matters that Roemer conceived the problem in 
terms of the Cartesian vortices; the point was that the circular 
motion of the terrestrial vortex should produce an apparent cur- 
vature of the path of light. In its modern form the problem 
was propounded and explained by Bradley, who discovered the 
effect of " aberration" in 1728. 

Huygens made the journey back to Paris in the middle of the 
summer of 1678. With him went Nicholas Hartsoeker, later 
known as a maker of lenses. Once more Huygens settled down to 


his old occupations. The period to which we have come was, 
unhappily, one during which he was not for long well. He was 
ill again in 1679 an d although he recovered he was compelled 
again to return to the Hague in 168 1 . From this last convalescence 
he never returned to Paris. We are consequently faced with the 
fact that this was Huygens's last stay in Paris, and one which was 
seriously interrupted by illness. 

Curiously enough, in view of the swift reverse which was in 
store, Huygens's prestige seems never to have been higher than 
it was at this time. It is clear that he was widely regarded as in a 
real sense the head of the Academic Royale des Sciences, the 
position of which seemed even more assured than that of the 
Royal Society at this time. The Royal Society in fact suffered 
from the political upheavals of the time and from the defection 
of some of its members. 1678 was the year of the Popish Plot, 
which, according to Titus Dates, aimed at the conquest of the 
kingdom by the Jesuits. As late as November it was held that 
" there hath been and still is a damnable and hellish plot, con- 
trived and carried on by popish recusants, for the assassinating 
and murdering of the King and rooting out and destroying the 
Protestant religion ". With the death of Oldenburg in this year, 
Huygens's relations with the society were practically at an end. 
The Academic Royale had on the other hand increased in 
vigour after its slow development in the first few years. Thanks 
to Colbert, Huygens and Auzout had been able to equip the 
Academic with all the laboratory and astronomical apparatus 
required and each year saw improvements in scientific technique. 
The society had begun to undertake enterprises such as the ex- 
pedition of Richer to Cayenne in 1672 and this had led to impor- 
tant information concerning the shape of the earth. 

Nevertheless, in France as in England there were jealous 
opponents of the new learning. The universities always feared 
an undermining of their authority if the scientific societies be- 
came too strong or too serious in their tasks, the Jesuits wished 
to have a monopoly of the new knowledge, and there were some 
who stirred the popular mind against investigations which 
seemed to a less and less degree to aim at the production of new 
inventions or the amelioration of life. In France, Paul Pelisson, 
who was writing a history of Louis's reign, gave Huygens space 
to deal with current criticisms and to enlighten the public on the 
aims and work of the Academic. 


In this review Huygens limited himself to the particular field 
in which he was an authority. He began by alluding to the need 
of astronomical studies and the great importance of the new 
observatory. The use of pendulum clocks and improved tele- 
scopes had made possible observations incomparably more exact 
and more easy than they were in the time of Tycho Brahe. The 
discovery of new stars, the confirmation of the ring of Saturn and 
the discovery of its satellites, a full study of the moon's surface 
and the description of comets and sunspots were among modern 
achievements. The discovery and measurement of the velocity of 
light were adduced as a consequence of such observations. Nor was 
such work without practical value : the occultations of Jupiter's 
satellites provided a method of determining longitudes, a problem 
for which the pendulum clocks might soon provide a better 
answer. The great appeal of the work of the Academic, however, 
lay in the steady expansion of man's knowledge and understand- 
ing of the world. Expeditions had been sent to Cayenne and to 
Hveen; more exact star catalogues and ephemerides were to be 
prepared so that the theories of the universe might accord more 
exactly with observation. The Earth itself had been made the 
object of scientific measurement. Geometry had been applied " in 
the study of causes in the field of Physics, it being accepted by 
almost all philosophers of today that the figure and movement 
of the corpuscles of which everything is composed are alone re- 
sponsible for all the wonderful effects which we see in nature ". 
This is really a statement of the new mechanistic philosophy to 
which the physical researches of the men of science had led. 
The world had come more and more to be regarded as a perfect 
machine and, says Burtt 1 , "first in Huygens and (in a more philo- 
sophical form) in Leibnitz we have this opinion unequivocally 
proclaimed ". Huygens clearly feared that this summary might 
seem to be written from the standpoint of the vortex theory of 
Descartes and he went on: " which opinion having been greatly 
supported through the philosophy of Descartes, they neverthe- 
less adhere neither to his sentiments nor to those of any other 
philosopher in order to gain authority ". Descartes, he pointed 
out, was mistaken in many things through lack of experiment 
and in particular he had sacrificed the accurate definitions of con- 
cepts which Galileo had begun to clarify. Truer ideas were now 
held regarding motion and force and momentum, the nature of 

1 E. Burtt: The Metaphysical Foundations of Modern Science (193*). 


meteors and other celestial phenomena, and the nature and 
effects of light. Microscopes, telescopes, the air pump and many 
other machines had been brought into use and had extended 
man's knowledge and led to the advancement of the sciences. 

Nevertheless, while Huygens was an enthusiast for the new 
world of the seventeenth century science, he displayed caution 
in comparing his own age with that of classical antiquity. His 
friend Charles Perrault frankly regarded the seventeenth cen- 
tury as superior to all other ages and considered Huygens him- 
self an example of this superiority. To such praise and to that 
of the younger Fermat, who compared him with Descartes, he 
returned a modest reply. " I am one of those who have profited 
from the wisdom of that great man," he wrote. 

That Huygens was not everywhere so popular and that there 
were factions in the Academic at this time can scarcely be 
doubted. The eminence of Huygens, in whose honour a medal 
was struck in 1679, was not agreeable to Cassini and de la Hire 
and the latter is known to have led an opposition to the entry of 
all foreigners and especially the friends of Huygens into the 
academy. In France just as in England religious differences were 
being exploited for political ends. The greatest division within 
the Academic Royale seems to have resulted not from national- 
istic or religious partisanship but arose between members who, 
like the original Montmorians, were eminently followers of Des- 
cartes, and those who, like Huygens and Mariotte, showed an 
increasing scepticism towards the Cartesian system. In these cir- 
cumstances Huygens seems to have felt more affinity with men 
of science who were not involved in the dispute and with Leibnitz 
in particular there grew up an interesting correspondence. 

It will be remembered that Leibnitz had studied mathematics 
with Huygens in 1672. During the subsequent years the German 
mathematician had pursued his researches along new paths. In 
1676 he had been in correspondence with Newton about methods 
of expansion in series. Newton mentioned his binomial theorem 
and the method of fluxions but did not describe the latter, 
although he added some illustrations of its use. By 1675 Leibnitz 
was employing his own form of differential calculus but was un- 
able to involve Newton in discussing anything which might 
arouse controversy. Newton's method of fluxions was, in fact, 
not published until 1693. In correspondence with Huygens, Leib- 
nitz claimed to have developed the calculus into a method by 


means of which he had successfully treated a variety of prob- 
lems. Huygens, however, would not abandon geometry for the 
differential calculus and never gained any facility in its use. Leib- 
nitz wrote at length also about the subject of symbolic logic, of 
which he was an originator, but his ideas were not appreciated by 
Huygens or anyone else at that time and they wer6 not taken 
up until the following century. Leibnitz was anxious to secure 
nomination to the Academic Royale as a foreign member but 
this Huygens seems to have been unable to obtain. Not until 
1700 was Leibnitz and in the same year Newton admitted as 
a foreign member. It was in 1700 that Leibnitz organized the 
Berlin Academy of Sciences. 

At the Academic Royale Huygens's chief activity at this 
time was the presentation of his work in geometrical optics in 
a series of lectures lasting from May to August. The whole sum- 
mer of 1679 was spent in editing the work of many years before 
and with special problems concerning Iceland spar. Much of 
the work on the optical properties of conic sections which comes 
at the end of the Traite de la Lumiere was completed about this 
time. Fermat's least-time principle, also, he succeeded in de- 
ducing for refraction on the assumption that light travels more 
slowly in glass or water than in air. As for Descartes, both 
Huygens and Leibnitz had scant regard for the greater part of 
his work in this field. His " pretence of a demonstration " of 
the laws of refraction was replaced by the well-known treatment 
which employs Huygens's secondary wavelets. Huygens's work 
in optics may in fact be regarded as standing in relation to 
previous studies by Kepler, Snell, Descartes, and Fermat much 
as Newton's mechanics stands in relation to the mechanics of 
Galileo and Huygens : Huygens achieved the same union of the 
physical and mathematical aspects of the subject. His mastery 
of geometry of course equipped him in a unique way for this 
task. The subject of colour was, however, left on one side; 
Huygens seems always to have held that a mathematical ex- 
planation of this was not possible. Nevertheless, he appreciated 
the practical outcome of Newton's work in this subject: the 
discovery of chromatic aberration of lenses showed, he saw, that 
this effect may be no less important in telescopes than spherical 
aberration. It followed that the search for the achromatic lens 
might be more profitable than the attempt to obtain lenses with 
non-spherical surfaces had been. He would probably have had to 


admit by this time that the idea of the reflecting telescope was 
by no means " extravagant ". 

The interest aroused in microscopic work by Hooke's Micro- 
graphia and the translation from Dutch into French of 
Leeuwenhoek's work by Huygens about 1677 led many at this 
time to take up such studies. The whole world of infusoria 
awaited discovery. The imperfections of the compound micro- 
scope were still considerable, however, and Leeuwenhoek, as is 
well known, preferred to use a single lens of short focal length in 
his observations, which perhaps included the discovery of bac- 
teria. Huygens used very small lenses of glass, some of which 
he made hollow and filled with alcohol. Locke, who was in Paris 
in 1678, wrote to Boyle of the " extraordinary goodness " of 
Huygens's microscopes. In devising a mount for his very small 
lenses Huygens introduced a method of altering the intensity of 
illumination of the object. Later, in 1692, he introduced dark 
ground illumination. These were the contributions of a prac- 
tical microscopist. After 1676, in fact, Huygens was very 
interested in making observations of infusoria in rain water. 

Ill-health no doubt accounts for a diminution of the mathe- 
matical and more abstract studies of Huygens after 1680. He left 
Paris at the end of the summer of this year for a short stay at 
Viry, where the country air restored him for a time. He returned 
to the capital in time to take part in observations of a comet and 
as a recreation started the construction of a planetary machine 
which would reproduce by means of clockwork the relative 
motions in the solar system. Early in 1681 he was again ill, but 
not until September was his return to Holland practicable. 


The convalescence after this last illness was slow. Letters 
arrived bearing the good wishes of men of science in Paris and 
London. Even de la Hire, only recently elected but before long a 
prominent member of the Academic, sent the good wishes of 
" all the company ". It is evident from his letter that de la Hire 
hoped for the position which Huygens had left at least tempor- 
arily vacant; there is a strong presumption that he did in fact 
work hard to prevent Huygens from having much opportunity 
to return to Paris. Huygens for his part was at first in no hurry 
to leave Holland even when, in 1682, he had practically 


recovered. Later in the year the Dutch East Indies Company 
showed interest in the latest pattern of marine clock and this 
was a further incentive to remain. By taking up the summer of 
1683, the work on the new marine clock really decided his future, 
for Colbert died in September of this year and without his 
patron's support the opposition to Huygens's return began to 
be really formidable. Not only this but the political horizon was 
dark with the uncertainties caused by the renewal of Louis's 
activities abroad. By 1686 the situation in Europe was not unlike 
that of 1673. The Revocation of the Edict of Nantes (1685) 
roused all Protestant countries. To Holland fled a large number 
of exiled Frenchmen. It was a time of rapidly diminishing free- 
dom in France and Huygens's experience well illustrates the fact. 
For when he renewed his application to return the appeal fell on 
deaf ears. Whether anti-Protestant feeling was the sole reason 
is not altogether clear. Huygens's friend Roemer left Paris a few 
months earlier, and it was four years later that the Edict was 
repealed. Quite as much as anti-Protestantism, probably, per- 
sonal jealousies spoiled the work of the Academie; the years 
after 1681 seem indeed to have been years of retrogression 

It is a striking illustration of the hostility which the 
Cartesians had come to feel for Huygens at this time that the 
Abb Catelan should, nine years after the publication of the 
Horologium Oscillatorium, attack the fundamental principles 
employed by Huygens in his treatment of the compound 
pendulum. There is nothing of scientific interest in Catelan's 
criticisms and they were designed to discredit Huygens's work 
in the eyes of those who were comparatively ignorant of mathe 
matics and mechanics. The mathematician James Bernoulli 
came forward to champion Huygens's ideas in 1684. 

In the middle of 1684 Huygens was visited by Thomas 
Molyneux, a contemporary and acquaintance of Flamsteed and 
Hooke. Molyneux wrote to his brother that he was received 
" extraordinarily civilly ". Huygens, he said, " beyond my ex- 
pectations talked to me in my own language, and pretty well ". 
He was shown Huygens's planetary machine which he decided 
was " nothing more than an ingenious curiosity " for, he said, " I 
asked him could he by help of it exactly determine an eclipse, 
and I observed that he could not give me a positive answer, as 
being loath to confess the imperfections of his contrivance to me 
that seemed to admire it so much as I did ". Huygens had in fact 


come to the end of his great period of scientific activity, but 
some profound ideas were yet to be developed. The news of the 
death of Picard (1682), who had done notable work at the Paris 
observatory, caused Huygens to ponder the uncertainty of life 
and to think of publishing the " many good and useful things 
which I have written or found, to complete which I desire only 
peace and the continuance of my health ". As events turned out 
his retirement held more of solitariness than he desired. The 
death of his father at a great age in 1687, and the departure of his 
brother Constantin for England in 1688, when William of 
Orange became King of England, left him alone in the isolated 
residence at Voorburg in the summer. The winters he spent at 
the Hague. In his letters he lamented the absence of any with 
whom he could discuss scientific topics. Owing to financial 
worries he began to consider the possibility of securing a position 
as counsellor to William III, but this only embarrassed the King, 
who perceived that Huygens had " higher ideas than to loiter 
with administrators ". 

The idea of a position in England seems to have occurred to 
Huygens after a short visit to this country in 1689. He was in 
London from June to August of this year, but only brief records 
remain. He met Flamsteed at Greenwich and attended a meet- 
ing of the Royal Society at Gresham College. In company with 
Fatio de Duillier, a Swiss mathematician, he met Newton for the 
first time. Little is known about this or of another occasion 
when, in July, Huygens, de Duillier, and Newton travelled from 
Cambridge together on the occasion of Newton's application 
for the position of Provost of King's College. Huygens also met 
Boyle on several occasions and witnessed some chemical experi- 
ments. He left London with many regrets for the isolation in 
which he then lived. 

It would, of course, be extremely interesting to know what 
discussions took place between Newton and Huygens on the 
occasions when they met. It is clear that in mechanics the two 
scientists held certain divergent views, notably on the subject of 
the conservation of energy and on the existence of absolute space 
and time. For Huygens, after reading the Principle, became 
strongly critical not only of Newton's postulate of universal 
gravity but also of his belief in the existence of absolute space 
and motion. He had early perceived that a body, moving uni- 
formly in a straight line with respect to one observer, might be 


accelerated with respect to another. And while he first made a 
distinction in favour of the absolute nature of motion in a circle 
which was accompanied by the existence of a centrifugal force, 
he abandoned this also after reading the Principia. This was in 
contrast with Newton's acceptance of an absolute space and time 
according to which all motion possessed an absolute character. 
Whether they discussed such differences of view and whether 
they compared notes on the subject of resisted motion and other 
matters in which they were both interested at this time is not 

Over the question of the cause of gravitational attraction 
there was, of course, a complete divergence of view between 
Newton and Huygens, for while the latter speculated for some 
time on the subject it was one for which Newton felt no interest. 
Huygens went out of his way to expound his views in such a way 
that they would not give Newton any offence. He seems to have 
been a little nervous about Newton's reception of yet another 
hypothesis. It may be remarked here that Huygens's theory 
grew out of his work on the nature of light and was an attempt 
to explain gravity as due to the action of an ether or fluid matter 
which, owing to rotation, seeks to travel away from the centre 
and thus, as he thought, forces slower moving bodies together. 
Even at this time, however, the difficulties of such a theory were 
becoming clearer. A fluid which could permeate matter could 
scarcely exert a reaction on it and de Duillier, who had gone to 
England for the purpose of studying Newton's works, pointed 
out to Huygens that the absence of any apparent resistance to 
the motion of planets and comets argued that the ether must be 
excessively attenuated. As is now known, however, Newton was 
not so thoroughly opposed to the ether theory as was generally 
supposed. Although he condemned the idea (as expressed by 
Hooke) in 1675, he returned to the question in the " Queries " to 
his Optics. 

The inverse-square law of gravitational force posed great 
difficulties for Huygens's mechanistic theories. It was, he said, 
" a new and very remarkable property of gravity of which it 
was very necessary to search out the reason ". He could not see 
that the cause could be given on the principles of mechanics or 
of the rules of motion. The view that gravity was an inherent 
property of matter, he said, "takes us very far from the 
principles of mathematics or mechanics ". Leibnitz also was 


against Newton's view of gravity as innate. If it was to be 
regarded as a " law of God who brings about this effect without 
using any intelligible means, then it is -a senseless occult 
property, which is so very occult that it can never be cleared up, 
even though a Spirit, not to say God himself, were endeavouring 
to explain it," he wrote to Hartsoeker. 

The importance of Newton's work was not, however, lost on 
Huygens, who perceived that the Principia utterly destroyed the 
Cartesian vortices. Writing to Leibnitz about the elliptical orbits 
of the planets Huygens said he would like to know if he could 
continue to hold to Descartes's vortices after reading the 
Principia since these " in my view are superfluous if one accepts 
the system of Mr. Newton in which the movement of the planets 
is explained by the gravity towards the Sun and the vis centri- 
fuga which balances it ... ". 

The extreme Cartesian view of gravity was expounded (1690) 
by Regis in a book on Richer's observations at Cayenne. The 
explanations put forward were closely similar to the ideas ex- 
pressed by Huygens in 1669 on the occasion of the discussion at 
the Acad&nie Royale. Regis made no mention of Newton in his 
book. In 1690 Huygens felt a good deal of uncertainty and 
wavered between his original ideas and the view expressed in his 
letter to Leibnitz. The appearance of the tract on the cause of 
gravity at the end of the Traitf de la Lumtire, published in this 
year, cannot be held to represent Huygens's final views, about 
which more will be said later. In England the effect of the 
Principia was more profound. Fatio de Duillier said that some 
of the Royal Society were "extremely prepossessed" in the 
book's favour and reproached those who were not under its spell 
as being too Cartesian. " They . . . have led me to understand 
that after the meditations of their author all Physics has been 
much changed " lie wrote to Huygens. There can be no doubt 
that on the Continent the criticisms made by Huygens and 
Leibnitz strengthened the position of the Cartesian philosophy 
for a good many more years. And yet Huygens's own work was, 
at its best, as opposed as Newton's to the Cartesian frame of 
mind and he did a great deal to dispose of the errors of 
Descartes's physical ideas, llie last five years of Huygens's life 
were in fact to be years of crisis for the Cartesian philosophy. 
Leibnitz and Huygens would have developed an alternative 
analysis which freed itself from Descartes's errors while at the 


same time rejecting Newton's conception of matter, time and 
space but this project was never carried through. In the event 
their effect was to delay the acceptance of Newton's work while 
at the same time weakening the supports on which Descartes's 
physical teaching rested. 

In the meantime the more orthodox Cartesians were driven to 
great lengths to show that the new results of scientific research 
were fundamentally in accord with Descartes's ideas. Even the 
differential calculus was claimed by Catelan to be derivable from 
Descartes's geometry. The faulty treatise which he produced to 
support his view led to a dispute with the mathematician de 
I'Hdpital. The latter, regarding Huygens as a natural ally, gave 
violent support to the latter's mechanics, also criticized by 
Catelan. This somewhat embarrassed Huygens, who was by no 
means sure of some of de I'Hopital's ideas on this subject. For 
de l'H6pital tried to obtain some sort of proof of the principle 
that the centre of gravity of a system of connected bodies cannot 
rise under the sole action of gravity. Huygens preferred to 
regard this principle as self-evident. Pascal and Torricelli, he 
pointed out, had used the same idea though limiting it to 

Of a different character was Huygens's correspondence with 
Pierre Bayle about this time. This famous French sceptic was 
appointed professor of philosophy at Rotterdam in 1681, so that 
he arrived in Holland in the year that Huygens returned from 
Paris. Under the conditions of Catholic intolerance the intellec- 
tual ferment, once concentrated in Paris, was becoming diffused 
into the freer but less educated provinces and into Holland. In 
1684 Bayle started a periodical entitled Nouvelles de la 
r^publique de lettres, the first number of which he sent to 
Huygens. The latter became interested in Bayle's aims and 
received him at his house, where he enlightened him on the 
subject of scientific studies. His correspondence with Bayle came 
to an end, however, after the philosopher was condemned as an 
atheist in 1693. Bayle's view was that religious dogma is of its 
nature irrational and that there is no merit in Relieving that 
which is merely consonant with reason. This outlook of credo 
quia absurdum was one which could not appeal strongly to 

Rather more interesting was Huygens's correspondence with 
Pierre Daniel Huet, another sceptic whose avowed purpose was 


to discredit reason in the realm of religious belief. Both Huet 
and Bayle actually influenced thought in the opposite direction 
to that which was intended, that is, towards scepticism. Huet in 
addition was strongly opposed to the rationalism of Descartes 
and strongly in favour of empiricism. In October 1689 he sent 
Huygens a copy of his Censures on the Cartesian philosophy. In 
reply Huygens said that he also had meted out rough treatment 
to Descartes, and that he hoped that his scientific work had 
replaced Descartes's doctrines with truer ideas. He agreed with 
Huet that while Descartes had overthrown the older philosophies 
he had borrowed from them their dogmatic spirit. He had had 
the ambition to be the author of a new philosophy and in his 
haste had been led to maintain ideas even against disproof. To 
Huygens this philosophy appeared as the successor of Aristotel- 
ianism. Nevertheless, when Martin van Helden, a Cartesian and 
professor of mathematics at Louvain, was threatened with im- 
prisonment for his criticisms of scholastic philosophy, Huygens 
assisted him so that he should not become "a martyr to 
Cartesianism ". He did not feel very strongly over this matter, for 
the battle of experimental science against the a priori philoso- 
phies seemed to him to be won. " It seems to me " Leibnitz was 
able to say, " that the Cartesians have very much declined and 
that they have not too many able men ". 


In 1685 Huygens was still negotiating over his return to Paris 
and there were endless letters, many of them unanswered* sent 
off from Holland. Nevertheless, it is not really certain that he 
wanted to go back, and he may well have been deterred by know- 
ledge of the changed conditions at the Academic. Up to 1688 
he stayed on at the Hague, and in the spring of that year he 
settled at Hofwijk, a property in the neighbourhood of the city 
which had belonged to his father. After his father's death in 
March 1687, the house was lent to Christian by his brother 
Constantin, who left with William III on his memorable expedi- 
tion to England in the following year. 

In these last years (1685-95) Leibnitz was solicitous about 
Huygens's unpublished works and recommended him to con- 
serve his strength, for, he wrote, " I do not know anyone who 
could replace you ". Huygens's old age was a lonely one and he 


was troubled with ill-health, but, he replied, " I see that one 
becomes accustomed to all these things ". He was not quite for- 
gotten, for Charles Perrault drew public attention to the great 
scientist's work in his Parnllele des Anciens et des Modernes and 
when Varignon was about to publish a book on mathematics he 
felt that he must take the opportunity, as he said, of paying 
homage to " the greatest mathematician of our age ". 

The pattern of Huygens 's life remained much the same right 
up to the end. He continued to work on the improvement of 
lenses, on the spring-regulated clock and the marine clock, and 
the writing of his last work, the Cosmotheoros. Undaunted by 
the unpromising performance of the various marine clocks he 
had constructed since 1663 he continued, with characteristic 
patience, to labour at this still urgent problem of the marine 
chronometer. In 1685 he went himself on a short trial (the only 
one) on the Zuyder Zee. In 1686 and 1690, clocks fitted with 
bifilar pendulums were sent in charge of the captains. All these 
trials were unsuccessful. The failure of the bifilar pendulum was 
the greatest disappointment, for work on this type of clock dates 
as far back as 1673 at least, since it was described in the 
Horologium Oscillatorium. After 1690 Huygens experimented 
with a new type of regulator and reverted to the spring drive 
which he had tried at the beginning and then abandoned. The 
new clock went well in laboratory trials and in 1694 Huygens 
hoped that the Dutch East Indies Company would take it up. He 
died before anything further could be done. 

The fact that Huygens could not accept the chief conclusions 
of Newton's Principia is the most interesting fact that comes out 
of his correspondence at this time. Five years after its appear- 
ance he wrote of Newton, " I esteem his understanding and 
subtlety highly, but I consider that they have been put to ill use 
in the greater part of this work, where the author studies things 
of little use or when he builds on the improbable principle of 
attraction." The idea of universal gravitation " appears to me 
absurd " he wrote. Yet he felt compelled to admit that Newton's 
explanation of comets was incomparably better than anything 
imagined by Descartes. It was difficult to see how comets could 
cut across the vortices imagined by Descartes, or to explain the 
eccentricity of the planetary orbits and the real accelerations and 
retardations of the planets in their orbits except on the lines laid 
down by Newton. Over the shape of the Earth, also, Huygens 


was in accord with Newton. He did not deny that if the gravita- 
tion of the planets towards the sun were taken as inversely 
proportional to the square of their distances " this, with the cen- 
trifiigal virtue gives the Eccentric Elliptics of Kepler ". But he 
and Leibnitz, far from feeling that this reduced the solar system 
to order, felt that it raised an insistent question of how gravita- 
tion arose. Leibnitz thought he could perceive an analogy with 
the intensity of light which, as a simple geometrical deduction, 
also obeyed the inverse-square law. Rays of attraction might be 
imagined which caused bodies to descend if their centrifugal 
force diminished. These rays were dismissed by Huygens, how- 
ever, as incompatible with his theory of a circulating medium. 
It almost seemed as if a return might be made to Kepler's identi- 
fication of gravity with a kind of magnetic attraction. Leibnitz, 
at least, inclined not a little to this view. Both he and Huygens 
insisted on attributing the effects of gravity to the medium 
which they believed pervaded the universe. Consequently they 
were both interested in the study of motion in a resisting 
medium, for they no doubt perceived that this was the Achilles' 
heel of their system. If the medium had mechanical properties 
exhibiting themselves as gravitational force, magnetic force and 
in other ways, what influence must it have on the orbital motions 
of the planets and on terrestrial motions? Newton's Principia 
had dealt with this problem and much of the work was deliber- 
ately aimed at the overthrow of the Cartesian vortices. Huygens 
considered Newton's treatment to be not without fault but he 
agreed with him as against Leibnitz over the definition of resist- 
ance, " for you," he wrote " call the resistance the velocity lost 
or the loss of velocity caused by the medium . . . For Mr. 
Newton and myself, however, the resistance is the pressure of the 
medium against the surface of the moving body . . . " It is 
really astonishing to us now that Huygens did not see that 
Newton's study of resisted motion completely disproved the 
vortex theory, but we must remember that the elastic fluid theory 
was held in the nineteenth century under even greater difficul- 
ties. Furthermore, a comparison of Huygens and Leibnitz at 
this date leads to a decision in favour of Huygens's notions. In 
1692 Leibnitz still supported vortices, while accepting Kepler's 
laws; Huygens had at least got to the point of seeing the over- 
whelming force of the quantitative work of Newton even while 
he rebelled against innate gravity. As he finally left them, the 


vortices were considerably attenuated affairs, suitable only for 
popular exposition of the sort found in the Cosmotheoros. 
Leibnitz, on the other hand, converted the subtle matter of 
Descartes into a production of his own : the materia ambiens. 

A good deal of what we should term pure mathematics 
crops up in Huygens's and Leibnitz's letters. Leibnitz took up 
several of the problems studied by Huygens and gave them 
new form. " My design has been " he wrote, " to give a little 
trouble to these good Cartesians who, through having read the 
Elements of Bartholin or Malebranche, believe they can do all 
in Analysis." There followed a series of letters in which Leibnitz 
gave Huygens an account of the differential calculus and its 
uses. He was able to investigate the properties of a curve like the 
cycloid, he said, from a purely analytical treatment and without 
any recourse to the figure. In regard to the calculus, Leibnitz 
was not a clear expositor. It is clear that from one aspect the new 
method was regarded not so much as a development of pure 
mathematics as an instrument for physical research. The union 
of mathematics with experiment is what Sir William Dampier 
has called the " new mathematical method ". For Huygens, as 
for Galileo and indeed for Newton, experiment had not achieved 
the position it later held in certain branches of science. From a 
comparatively few observations, by the aid of " geometry " one 
could advance far into new realms, a fact which is well illustrated 
by Huygens's work on impact and on the compound pendulum. 
" It must be admitted " wrote Huygens, " that geometry is not 
made for all sorts of minds." 

From being sceptical Huygens soon became envious of the 
calculus differentialis. Finding Liebnitz's accounts rather 
obscure he wished that either he or Bernoulli could be there to 
assist him. Some collaboration did indeed spring up with Fatio 
de Duillier, and Huygens's note-books contain many pages of 
working on the new lines. The great change of outlook was a 
difficult one for the great geometer and he did not attain facility 
in the use of the calculus. The new calculus, Leibnitz 
emphasized, gave its results by a kind of analysis without any 
effort of the imagination, " and it gives us over Archimedes all 
the advantages which Vieta and Descartes have given us over 
Apollonius ". 

Fatio's work on the calculus is important in the history of the 
subject, for it was through him that the dispute between the 


followers of Newton and Leibnitz sprang up. More's Isaac 
Newton (1934) gives a good account of the episode. Fatio seems 
to have become resentful of Leibnitz's rather superior criticism 
of his work, which was of an undistinguished nature, and it is 
considered that Fatio smarted under a sense of grievance. After 
returning from England, where he had been in contact with 
Newton, Fatio wrote to Huygens, saying that priority for the 
invention of the differential calculus certainly belonged to 
Newton. He suggested that Leibnitz's ideas were in fact obtained 
from Newton's letters which went back to 1676 and 1677. The 
publication of these, he hinted, would embarrass Leibnitz. 
Actually, these dates were beside the point since both mathe- 
maticians seem to have used the calculus much earlier. Leibnitz, 
at any rate, affected to be unimpressed by news of Newton's 
advances and hinted to Huygens that he had done a number of 
things of which Newton knew nothing. Huygens's part in the 
quarrel which ensued was small but noteworthy for it was 
through him that Leibnitz first learned of the charges made by 
de Duillier. 

About this time an interesting comparison in mathematical 
methods was made through the study of the same problem by 
Leibnitz, Huygens and James Bernoulli. This problem was the 
one propounded by Mersenne many years before; to find the 
theoretical form of a chain suspended from its two ends which 
are at the same height, so that a curved line hangs between them. 
The publication of the results showed a fair agreement between 
the three mathematicians but showed up the advantages of the 
calculus, which was more and more applied, not only to new 
problems but also to others already solved by classical methods. 
Huygens was not altogether pieased by some of the new 
methods, notably that adopted by James Bernoulli in solving the 
problem of the centre of oscillation. But the truth is that the 
form of Huygens's work in mathematics had always been some- 
Wjhat reactionary and was fast becoming obsolete. 

There is a faint echo of the Royal Society and the pleasant 
days spent in England in some of Huygens's last correspondence. 
Constantin, his brother, in 1691 presented a telescope objective 
having a focal length of 1-22 feet to the society. This was by no 
means the best achieved by Huygens. During these years he 
made one of 210 feet focal length. The 122 foot lens was a fine 
objective, however, and Hooke was entrusted with the erection 


of an aerial telescope to accommodate it. The lens was marked 
by Constantin so that it should afterwards be readily identified 
Constantin apparently not finding Hooke any more trustworthy 
on personal acquaintance than he had anticipated. This mark- 
ing has made possible the identification of the objective and 
examination of its quality in recent times. The figuring and 
centring of the surfaces is described as "astonishingly efficient," 
but the quality of the glass is " hopelessly bad ". From 
Constantin, Huygens heard of the death of Robert Boyle. " Mr. 
Boyle is dead as you know already without doubt " he wrote to 
Leibnitz, and added " It seems strange that he built nothing on 
all the experiments with which his books are filled; but the thing 
is difficult, and I never thought him capable of as great an 
application as is necessary to establish the true principles." 
Since Huygens had often expressed his admiration of Boyle this 
comment may be taken to indicate more esteem of the difficul- 
ties of Chemistry than depreciation of the father of that science. 

The other prevailing practical interest, besides the clock, to 
which Huygens gave attention during these last years was the 
telescope. In 1684 he published his Astroscopia Compendiaria. 
This contained an account of tubeless telescopes and this may 
explain how it is that Huygens has been supposed to be the 
originator of this difficult and ultimately unsatisfactory tech- 
nique. He was compelled to reconsider his attitude to the reflect- 
ing telescope through the apparently insuperable obstacles raised 
by aberration chromatic and spherical. Yet he still preferred 
the refracting telescope because metal mirrors took such a poor 
polish and the grinding of glass mirrors was extremely difficult. 
Moreover, it was at that time very difficult to silver the upper 
surface and silvering the back surface meant that a second image 
was formed by partial reflection at the top surface so that a 
double image was formed. Even if the radii of curvature of the 
two surfaces were such that the two images were coincident the 
difficulty remained that the images could not be of the same size. 
Huygens's conclusion that in practice the weaker image would 
not be distinguished by the eye if the images were coincident 
led him to renewed experiments but the work does not seem to 
have progressed very far. 

To these last years belong the remainder of his work on his 
Dioptrica, a paper on harmonics, his continued studies in mathe- 
matics partly in collaboration with David Gregory, who visited 


him in 1693 and the clarification of certain points in his atomic 
theory. Huygens was a strong supporter of the atomic doctrine 
which was later to provide the basis of a scientific development 
of Chemistry, but he is not usually numbered among those who 
developed the atomic theory simply because his researches led 
him in the direction of Physics. He continued to oppose Newton 
over the doctrine of universal gravitation and added to this his 
difference from the English scientist over the corpuscular theory 
of light then gaining wider acceptance. The extreme rarity and 
speed of the corpuscles and the absence of any obvious way of 
explaining colour on this hypothesis were Huygens's main objec- 
tions to the theory which was for a century to overshadow his 
own elegant treatment of the propagation of light. Huygens 
however, was not inclined to seek controversy. In 1694 he had 
occasion to correct Renau, an engineer to Louis XIV, over his 
connotation of force, a matter which was too important to over- 
look even in his declining health. 

Huygens himself recognized that the illness which had 
dogged him since his return from Paris had become much more 
threatening. In March 1695, Huygens felt it necessary to call his 
lawyer and make the final corrections to his will. The following 
month he became worse and from then until July, pain and 
sleeplessness spared him hardly at all. He lived in dread of 
losing his reason and his days were filled with a deep despair. 
Constantin visited him for a few days at the end of May. Neither 
he nor the Calvinist pastor who was at last brought against 
Huygens's wishes could give him comfort. Against the doctrines 
of personal immortality and the exhortations of the Reformed 
Church he maintained an obstinate scepticism. Huygens died 
facing the problem of individual and personal immortality as " a 
problematical question/' his attitude in this contrasting with that 
of other seventeenth century scientists, especially Boyle, Pascal 
and Newton. 

Nevertheless, when the end came during the afternoon of 
July gth, the failing spirit, weakened by suffering, may have 
found some serenity. The author of Cosmotheoros, it has been 
well remarked, revealed himself, in the face of problems less 
abstract than those of mathematics and mechanics, as sensitive 
to the deep realities of life and the profound aspirations of the 
human spirit. But he could not forget that all experience may be 
subjected to the scrutiny of a dispassionate mind, and dispa* 


sionate and remote Huygens always seems to have been. He 
lacked perhaps, a sense of the mystical and preferred to look on 
life and death with the vision of one to whom all things 
appeared capable of an ultimate rational explanation. For him 
as for Mersenne, the technique and method of the sciences were 
exalted because they freed men's minds from error. But for 
Huygens there was no " two-fold truth". Truth for faith and 
truth for reason were, for him, one. 

The professional and serious interests of Huygens are the 
ones which are uppermost in his correspondence. Nevertheless, 
it would be a mistake to consider him as always having been 
nothing but a patient researcher. He was a man of wide culture 
and of acquaintance throughout Europe. The poetry and music 
of his own country, with which he was well conversant, were of 
a high order; its painting reached the summits marked by the 
names of Rembrandt, Franz Hals, Vermeer. In Paris, Huygens 
used frequently to visit noted musicians, among them the 
harpsichordist Chambonniere. He himself played the harpsi- 
chord. Nor was he averse to feminine society. One reads of his 
meetings with Marie Perriquet at the country house of the 
academist Conrart, with fashionable women at the lectures given 
by Rohault. Still more interesting is the fact that he was an 
occasional visitor to the salon of Madelleine de Scudery. 
Marianne Petit, daughter of one of Louis XIV's engineers, seems 
to have had especial charms for Huygens but she never married 
and their separation was due to her withdrawal from society 
when she entered a religious order. There were even scandals 
about Huygens during the war of 1672-8, for he paid frequent 
visits to Madame Buat, a grand-daughter of the Dutch poet 
Cats, but not possessed of the highest reputation for virtue. 
There were, too, some distant cousins whom he visited in Paris, 
and for the eldest of these there is no doubt he felt considerable 

Huygens moved in an elegant and leisured society during his 
years in Paris and there are echoes of days at Viry, where the 
Perraults had a fine country house, as well as faint glimpses of an 
elegant Huygens visiting occasionally the salons of the city. 
The contrast of this varied life with the quiet of his days near 
the little village of Voorburg must have been marked. But in 
fact great changes were taking place and Catholic intolerance in 
France was undoing much that Colbert had been at pains to 


build. After 1685, lt was impossible for Huygens to think of 
returning to Paris. Holland remained what she had been early in 
the century, a refuge for free spirits, a country where new 
thoughts might still be brought into the world. But Huygens 
was past the days of his vigour and little that is new can be 
attributed to his later years. As an exceptionally placed observer 
he had seen a good deal, moving as he had done, in the diplo- 
matic and learned worlds of his time. The political absolutism 
of the age must have contrasted strongly, one would have 
thought, with the new freedom of speculation which survived all 
repressive measures. But Huygens does not appear to have 
remarked on this. The doctrines of expediency, sovereign right, 
vital interest and the like have survived, and freedom of thought 
requires to be continually defended. That much is clear. But 
Huygens belonged really to the world of abstract thought 
and shrank from contact with political affairs. In his own world 
he was as Leibnitz and John Bernoulli agreed, " incomparable ". 
His loss, Leibnitz averred, was "inestimable" for he had equalled 
the achievements of Galileo and Descartes; helped by what they 
had done he had surpassed their discoveries: " In a word he 
was one of the greatest ornaments of this time." 


The State of Science in the First Half of the Seventeenth 


THE task of this chapter is not so much to describe actual 
achievements of the men of science as to explain how at this 
period some of them looked at Nature, and to indicate what 
particular problems they felt called upon to solve. Such a ques- 
tion bristles with difficulties. It may be that it is of the type 
which beckons on only the foolhardy. However that may be, it 
certainly must be admitted that the views expressed are neces- 
sarily of rather a personal character; they will invite, on some 
points at least, as much dissension as agreement. 

Very probably there is a good deal to be said for the view 
that steady progress in the physical sciences had to wait until 
the seventeenth century because of the unsuitable manner of 
analyzing events which came from the study of Aristotle. Aris- 
totle's interest, undoubtedly, lay rather in logical relations and 
his notion of change fitted in with the conception of develop- 
ment of form, of growth of living organisms, far better than it 
did with the phenomena of the inanimate world. Rejecting the 
notion of sheer physical determinism somewhat as Plato had 
done, he directed all attention to the purpose served by events 
and hardly any to the conditions which were invariable ante- 
cedents. The question why things happened as they did was 
always tempting and appealed to men as an outlet for tempera- 
ment; it was so much less restrained and impersonal than the 
question how things are caused, and so long as men's minds 
were bemused with an elaborate hierarchical scheme the latter 
scarcely seemed to matter. For Aristotle held that change is 
purely a terrestrial affair, at least it must be limited to the sub* 
c 97 


lunary sphere, and within that sphere there were just a few 
natural motions through which the different elements could 
attain their proper level. The natural motion of " earth " down- 
wards, or of " fire " upwards, were thus matters which invited 
no discussion. For " unnatural " motion, such as that of a 
lump of earth horizontally, a constant push must be applied. For 
motion in a circle, on the other hand, as in the case of the 
apparent motions of the heavenly bodies, no acceleration to- 
wards the centre was ever dreamed of; Aristotle had nothing 
but reverence for this form of motion for it was, he taught, the 
perfect form, and proper only to the celestial bodies. 

Galileo's breaking away from scholastic doctrines must have 
come after a period of perplexity, for he no doubt tried out cur- 
rent explanations for what they were worth. From Roger Bacon 
to the Accademia del Cimento is a long distance in time and 
there was a considerable change in outlook. The difference was 
that the Accademia existed to learn about the world through ex- 
periment and to cast its explanations in a form which was no 
longer subservient to religion. But in such an intellectual climate 
as existed at the very beginning of the seventeenth century the 
only safe plan was to concentrate on the description of motion in 
geometrical terms and this, of course, Kepler was fortunate 
enough to do. Kepler stood little danger of the condemnation 
meted out to Galileo in spite of his almost equally great fame, 
because his work could be regarded as a presentation of Aristotle's 
formal cause in an elegant mathematical form. Mathematics, in- 
deed, remained respectable, supported still by the prestige of 
Plato, and in the sixteenth century there was in some of the Italian 
centres of learning a renewed interest in the teachings of Pytha- 
goras. From Pythagoras, Whitehead has remarked, " mathema- 
tics and mathematical physics took their rise. He discovered the 
importance of dealing with abstractions; and in particular 
directed attention to number as characterizing the periodicities 
of notes of music. The importance of the abstract idea of periodi- 
city was thus present at the very beginning both of mathematics 
and of European philosophy." For Kepler " the cause of the 
observed effects is the mathematical beauty and harmony dis- 
coverable in them ". 

The history of this mathematical treatment of phenomena 
is usually somewhat briefly dismissed, as if it all resulted from the 
insight of Galileo and Newton and all the rest were fools. But 


Copernicus, it should be remembered, showed a fundamentally 
similar attitude much earlier, although neither he nor Kepler con- 
cerned themselves with the physics of their problems. By 1543, 
the date of his De Revolutionibus Orbium Codestiwn, he had 
analyzed linear simple harmonic motion as the product of two 
combined circular motions. This is the first record of a study 
of such a form of motion which Copernicus himself described 
as pendentibus similes after the manner of suspended bodies. 
But, characteristically enough, circular motions were still 
thought of as raising no particular problems. Benedetti, how- 
ever, in his Disputationes of 1585, showed an advance in that he 
adduced linear simple harmonic motion in controverting some 
of Aristotle's ideas about motion. Then, as against Aristotle's 
teaching that bodies fall towards the centre of the earth because 
that is their " natural " place, he asserted, though he did not 
prove, that if a weight were dropped through a hole passing 
through the centre of the earth, it would by no means come to 
a stop but would oscillate backwards and forwards in the manner 
of a pendulum bob. This interest in accelerated motion marked 
the beginning of the criticism of Aristotle which was to grow 
so much in intensity during the next hundred years. It may 
indeed be said that modern science begins with the study of 
dynamics and in particular with perplexities about simple har- 
monic motion. 

Galileo, in his Dialogue concerning the Two Chief Systems 
of the World (1632), made considerable use of the simple pendu- 
lum in his study of naturally accelerated motion. He restated 
Benedetti's view of the subsequent motion of a body dropped 
through a hole passing through the centre of the earth. Dis- 
satisfied with the formal cause which set such limitations to Kep 
ler's work, he looked, in the case of accelerated motion, for what 
Newton later called an external impressed force. He even had 
some notion of the necessity of postulating action and reaction, 
for in writing of the simple pendulum he perceived that the 
thread, in so far as it possesses mass, must "hinder" the motion of 
the bob and that there was a problem the problem of the com- 
pound pendulum which remained to be solved. His study of the 
simple pendulum shows, however, a necessary amount of idealiza- 
tion, without which a relation between the period and the length 
would have been excessively hard to find. Galileo's work, in fact, 
marks the first successful union of experiment and mathematics, 


for he brought together the criteria of the simplicity of natural 
laws and their accordance with experiment. The problems 
he bequeathed were as important as those he solved: the 
concepts of mass and momentum, and of force and work, and the 
law now known as Newton's Third Law were all apprehended in 
only a somewhat confused way at Galileo's death. Mersenne's 
Hannome Universelle (1636) is not regarded as being of such 
moment as the writings of Galileo, to which Mersenne was in fact 
greatly indebted. He deserves credit, however, for being the first 
man in Europe to perceive Galileo's genius and to promulgate 
his ideas. 

So far all that has been said has been concerned with the 
bright and positive side of the picture of the early seventeenth 
century, but there was a vast difference between the conditions 
which then ruled and those of today. Today we are accustomed 
to an intellectual demarcation of Science and Philosophy and 
to almost equally sharp boundaries between the various sciences. 
Then there was a chaotic muddle of " philosophies " and no tacit 
agreement on the fundamental concepts through which phenom- 
ena are ultimately explained. If we single out the great 
names it is possible to see the seventeenth century as a period of 
amazing clarification, but this it must rarely have appeared to 
be to the contemporary men of science. The alchemists, for 
example, were engaged in the most obscurantist practices, and if 
Science as a whole had depended on their progress in their " art " 
the period would have to be written off as a failure. The works 
of Paracelsus, Basil Valentine and Raymond Lully which con- 
tinued to circulate, and the extensive forgotten literature pro- 
duced by a host of enthusiastic and credulous amateurs, take us 
back to the worst periods of superstition and it is simply not true 
that the dawn of the seventeenth century brought a universal 
belief in the operation of physical law and a discredit of 
magic and mystic agents. There was this one brilliant triumph 
of mechanics and theoretical astronomy, but so acute a thinker 
as Descartes missed the true answer to the sceptical questions he 
set himself. By falling back on an intuitional method he was no 
doubt able to shorten his estimate of the labour required in this 
way to produce a complete science of inanimate nature, but this 
was in essence the same mistake as Aristotle's. Only those with 
a weakness for verbal and logical classification could follow Des- 
caries willingly through his opening premises and voyage thence 


through the world he constructed. There was, of course, the 
enormous difference that Descartes 's universe was supposed to be 
governed throughout by physical laws, even though these hap- 
pen not to be the ones which actually operate. But the idea was 
there, and the conception was one of the universal mechanical 
operation of law. 

By comparison, the atomists who followed the teaching of 
Gassendi seem to be intellectually children, and the atomic doc- 
trine, by 1650, was not noticeably an improvement on the original 
version of Lucretius. Its adoption by Hobbes, a friend of Gas- 
sendi, and the most uncompromising materialist of them all, 
was a misfortune, for Hobbes's reputation with the men of science 
became such that he had only to support a theory and it was 
damned. Much the most important contribution to atomic 
theory came in fact from Boyle, and we may see in certain brief 
essays in this style an embryonic form of the kinetic theory. 

But the existence of so many principles of explanation, drawn 
from Aristotle, Pythagoras, Lucretius, Descartes and Galileo, and 
the inability of some men of science to distinguish what was 
fact from what was mere hypothesis, all this made for unneces- 
sary obscurity. The discipline which characterized mathematics 
and astronomy had somehow to be introduced into a field which 
too easily rioted into thickets of explanatory hypothesis. Refer- 
ence to authority became prohibited; an anti-intellectualist atti- 
tude had to be encouraged and we find that William Gilbert, for 
example, denounced the " vast ocean of books " produced by 
writers whose explanations exhibited merely verbal ingenuity. 
" Neither Greek arguments nor Greek words can assist in finding 
truth/' he added. True, Francis Bacon wrote in a similar vein, 
but to a less extent than Gilbert was he able to explain the 
method which should be employed; indeed, he hindered progress 
by his persuasive suggestion that there was one particular method, 
and especially when he argued for classification of facts and 
omitted to consider measurement. Nevertheless, the men of 
science succeeded as much by practical ingenuity as by abstract 
reasoning in reducing their problems to some sort of order, and 
by the time the Paris and London societies came into existence 
some confidence was felt that natural philosophy could and 
should contribute in no small degree to the material welfare of 
men. That natural philosophy might profoundly alter men's be- 
liefs about the world was not then thought likely, and it seems 


that this modesty on the more intellectual aspect was genuine. 
The Cartesian scheme went as far as the educated man could 
desire in showing how the claims of Biblical cosmology and 
natural philosophy might be reconciled. And up to 1670 it does 
not appear that fundamental researches were planned with any 
expectation of a new conception of the universe, doubtful though 
Huygens and others might be of the validity of some of Des- 
cartes's theories when examined in detail. Quite a large propor- 
tion of the scientific work going on in the i66o's was concerned 
with practical things : mining, navigation, military science, tex- 
tiles and so on. Following Bacon, the men of science seemed 
to believe that "the real and legitimate good of the sciences 
is the endowment of human life with new inventions and 
order ". 

There were good reasons for this practical bias, for many of 
the techniques of civilization had advanced relatively little over 
a period of centuries. With the expansion of commerce, for 
example, navigation particularly remained difficult owing to 
the lack of a reliable and practicable means of determining long- 
itude at sea; the geographers and cartographers had in this 
respect also gained little since the twelfth century. Hence it was 
that when Galileo discovered Jupiter's moons quite as much 
interest was attached to their use as a source of standard time as 
to their cosmological significance, representing, as they appeared 
to do, a solar system in parvo. Not until 1668, however, were 
ephemerides for the moons published; then their publication by 
Cassini gained for the astronomer the recognition of an invita- 
tion to Paris. The secret of longitude determination could clearly 
mean a great deal for the national income, for ships had to 
struggle on their voyages to the East Indies by means of dead 
reckoning and such checks of longitude as could be made by 
observation of landmarks. Bond tried to use magnetic isogonals 
as an answer to the problem and his book The Longitude 
Found (1676) was considered to be important by Halley, though 
nothing came of the method in practice. 

Astronomy remained the key to many of the problems which 
were felt to be urgent during the first half of the century and 
it was the science which was most advanced through the work of 
Tycbo Brahe, Kepler, Galileo and others. The invention of 
logarithms by Napier in 1614 was timely for, in Laplace's words, 
it doubled the lifetime of astronomers though Napier's labours 


were greatly inspired by the aim of proving the Pope to be Anti- 
christ. So soon as money could be obtained, the French decided, 
an observatory should be constructed at Paris. It must be remem- 
bered that as yet a convincing decision between the Ptolemaic, 
Tychonian, and Copernican systems had still to be given; even 
though the first was no longer seriously supported by men of 
science, it shared with Tycho Brahe's system the advantage that 
no observation of stellar parallax had ever been made. The sup- 
porters of Copernicus could attribute the absence of parallax 
only to the smallness of the effect and this depended on the 
dimensions of the solar system being very great. 

When, accordingly, as a result of the energy of Auzout, 
Richer, Picard and others, an expedition was sent to Cayenne in 
order to make simultaneous observations of Mars there and at 
Paris, the size of the solar system was the primary question in 
view. Richer at Cayenne and Cassini at Paris took simultaneous 
measurements of the altitude of Mars and in 1677 the planet 
was at its nearest to the earth. Knowing the base d of the triangle 
in the figure, and the base angles % l and 9 2 , the distance of the 
planet was easily computed. 
By a straightforward appli- 
cation of Kepler's laws of 
planetary motion all the 
required distances followed 
from a single determination 
of the distance of the two 
planets. For the sidereal 
times of the orbits were 
known and these were re- 
lated with the mean dia- 
meters. Kepler's brilliant 
work on the orbit of Mars 
and this, triumph of 1672 
thus alike rested on the 
patient labour of Tycho Brahe and the whole was a sermon 
to the men of science on the power of quantitative 
mathematical relations. The conclusion that the sun's mean 
distance from the earth was 87,000,000 miles was of the right 
order. As J. W. Olmstead remarks in I sis (1942), writing of the 
expedition, " The great consequence of the expedition for con- 
temporaries was, of course, the revelation of the tremendous 



dimensions of the solar system, as well as the prodigious size of 
the sun and some of the planets The disclosure, with some cer- 
tainty, of the gigantic distances and masses involved was, for the 
general public, almost overpowering." The old Aristotelian con- 
ceptions of the universe might linger on in some minds but 
these were shattering blows and it was inevitable that educated 
men of the period should accommodate the increasing know- 
ledge of Nature, up to 1672, with the only other plausible " sys- 
tem " in existence : that of Descartes. 

It may be true that, as Professor Andrade has remarked in 
Nature (1942), " The Cartesian scheme was easy, pictorial, gen- 
eral : the Newtonian difficult, mathematical, precise. " In such 
an opposition it may appear that Descartes's ideas were woolly 
and essentially unscientific. Nevertheless, it does not do to under- 
estimate the amount of preparation which preceded the New- 
tonian synthesis or to despise investigations of men whose minds 
were, at least tentatively, inclined towards Descartes's theories. 
However deplorably qualitative his theorizing, Descartes inspired 
a generation at least with a vision of Nature as a sphere of 
universal law embracing planetary motion, rainbows, the pro- 
perties of magnets and of lenses, all that could be explained 
by reference to underlying mechanisms. There was no 
limitation of the field of scientific investigations, and, more- 
over, the scientific explanations Descartes looked for were to 
possess an admirable economy of principles, the subtly matter 
and ether being trusted to account for a variety of phenomena. 
It was his desire for a pictorial mechanism which was 
the greatest weakness of Cartesianism, and if it had proved 
possible to reconcile his ideas with measurement and so formu- 
late mathematical relationships, there is no doubt that Descartes 
would have done this. But he was willing to sacrifice the world 
of observation if need be, and he did not see that there was 
a radical difference between Galileo's idealization of his 
experimental studies and his own bold embarking on pure 

All this became clear enough to Huygens in the course of his 
life; but the problems he took up can frequently be understood 
best by reference to Descartes's essays on the same subjects : the 
laws of impact, centrifugal force, the centre of oscillation, the 
behaviour of lenses, the nature of light and the cause of gravity, 
these are all topics discussed by Descartes and in a manner which 


serves to introduce Huygens's own exact investigations. Professor 
Andrade makes an excellent remark when he says : "What New- 
ton ignores is what Aristotle and Descartes tried to start with ", 
and his summary of Newton's achievement cannot be ques- 
tioned : " If we are asked to state in *a sentence what was the 
main effect of Newton's work on the thought of his time, I think 
that the answer must be that it was to establish the power and 
universality of the methods of quantitative science." But ever 
since the time of D'Alembert's "Preliminary Discourse " to the 
Encyclopedic of 1751 there has been a tendency to admire New- 
ton's magnificent work by depreciating the earlier work of the 
century, and especially everything which could be labelled Car- 
tesian. It is important to realize what useful work was in fact 
done by the men of this period, and when we come to a con- 
sideration of Huygens's work much allowance has to be made 
for the late appearance of his works as publications. Otherwise 
the effect outside the Academic might have been far greater. 
" The evidence shows," wrote Professor Harcourt Brown, 1 " that 
Huygens took up the best elements of Cartesianism, adding to it 
along certain lines, rejecting some of it in other directions, and 
that he followed whatever seetned most fruitful of the sugges- 
tions received from parallel studies in England, Italy and else- 
where. He counts as one of a group whose efforts prepare the 
modern age of science, ushered in by way of the encyclopaedism 
of the eighteenth century, whose urbane and moderate scepti- 
cism is a definite premonition of the century of Voltaire." 

It must be said that where Huygens's views most foreshadow 
the eighteenth century was in his acceptance of complete physi- 
cal determinism, even though he chose to cast this in rather rigid 
mechanistic form. This chapter may, therefore, end with a brief 
discussion of the mechanistic hypothesis of which Huygens was 
so strong an exponent. The influence of Gassendi may be seen 
from time to time, even though Descartes's system left little scope 
for the development of Gassendi's atomic doctrines and the two 
philosophers were in some senses regarded as being in direct 
opposition. But Gassendi seems to have strengthened in Huy- 
gens a leaning towards the thoroughly mechanistic view of 
Nature. Kepler had supposed that magnetic attraction resulted 
from a " mutual affection "; Gassendi considered it was a purely 
physical force. For him light was a material substance; there 
* Annals of Science, 1936. 


were indeed atoms of light and heat and for the propagation of 
sound an example of a doctrine carried to extremes. Such views 
were opposed to Descartes's conceptions, for the latter held that 
space or " extension " is not infinitely divisible. Both considered 
that the universe should be regarded as a machine, although in 
the last resort Gassendi seems not to have thought it regulated 
itself. It was a scientific form of this teaching which came out 
of Huygens's mechanical studies, although he himself preferred 
to observe a separation between science and philosophy. Cer- 
tainly, so far as they went, Huygens's mechanical explanations 
showed several radical differences from those associated with 
the name of Newton. In particular, Huygens found great diffi- 
culty in making universal gravitation a part of his scheme, pre- 
ferring rather to regard matter as itself quite inert and incapable 
of any action except through collision with other matter. This 
difficulty meant in practice that he remained a Cartesian. In 
1646 Descartes had ridiculed the " absurd belief " of Roberval 
that there could be mutual gravitation between lumps of matter. 
It would mean, he said, that a particle of matter was possessed 
of a soul and endowed with consciousness, so that it could know 
what happened across space and could in some occult manner 
exert its own influence there. For a Cartesian the really urgent 
problem was rather to discover the laws governing the transfer 
of momentum on impact and this, as we have seen, was one of 
Huygens's earliest studies. 

It seems quite certain, therefore, that some of Huygens's 
rather curious conceptions come from Descartes. To the reader 
of Descartes's Principia Philosophise, for example, there would 
appear to be no incongruity in treating light and mechanics as 
related subjects. Descartes, and following him, Huygens, 
thought to find the explanation of gravity and light in the pro- 
perties of media which would enable bodies to act on each other 
across intervening space. Huygens perhaps did for a time 
seek to bring in that mathematical #nd scientific form of Des- 
cartes's theories which Paul Mouy supposed that he actually 
created *. But in fact, while he was at the very beginning of his 
work, Huygens saw that there was exceedingly little truth in 
many of Descartes's scientific " principles ". On the other hand, 
he could not adopt Newton's attitude, which left the question of 
pictorial mechanism out of consideration to a very large extent. 
1 Le Ddveloppement de la Physique Caftdsienne (1934). 


As has been mentioned, Huygens considered that the conception 
of a gravitational force, innate and inherent in matter, appeared 
to be absurd and one which " takes us very far from the prin- 
ciples of Mathematics or Mechanics ". As an enthusiast for a 
kinetic theory of all change he could allow matter no properties 
save that of inertia. It has to be remembered that he considered 
he had obtained direct evidence for the existence of a subtle mat- 
ter. But for Cartesianism as a philosophic system he had little 
patience and he accordingly had several clashes with some of the 
Jesuits and doctrinaire Cartesians. When Kircher adopted the 
cosmology of Tycho Brahe, Huygens criticized him for what 
appeared to be nothing but timidity. " We others ", he asserted, 
are " without fear ". 

In a sense then, Huygens's support of some of Descartes's 
ideas up to so late a date in the seventeenth century must be 
described as an experiment in failure. Very probably it was an 
experiment that had to be made; its failure showed clearly the 
great limitations of the extension of the notions of sense in the 
field of theory. It may even have been the feeling of dissatis- 
faction with the position men like Huygens were reaching that 
drove Newton to make his great renovation of natural 
philosophy. The progress of scientific explanation may then be 
seen as a process of leaving out of redundant elements until one 
arrives at the really successful procedure of abstraction. If so, 
this was a return to the methods of Galileo. 

The practical value of much of Huygens's laboratory work 
needs no stressing, nor does his interest in the clock, telescope, 
microscope, and other experimental and observational aids need 
explanation. The experiments of Galileo and the Florentine 
academicians were done with the simplest of apparatus. But new 
discoveries, it was realized, often attended the use of new appara- 
tus, a fact which was well illustrated by the invention of the 
vacuum pump. The rate of progress was for some time in direct 
relation with the experimental resources available. Here theory 
and practice interlock. It was not possible to improve the tele- 
scope, to invent a new eyepiece, without a quantitative study 
of the defects of the ordinary lens. Problems of theoretical in- 
terest led to the construction of the vacuum pump; experiments 
with this instrument led in their turn to new theoretical specula- 
tions regarding the nature of air, and, in Huygens's case, the 
cause of gravity. One can scarcely estimate the extent to which 


new and improved instruments assisted the great scientific 
advances of the seventeenth century. The telescope, microscope, 
barometer, vacuum pump, pendulum clock, micrometer eyepiece, 
and other instruments might be mentioned here. Without them 
rapid progress would have been unthinkable. 


Work on Collision between Elastic Bodies 

The Cartesian mechanism was governed by the principles of 
inertia and of the conservation of motion, the laws of impact, the 
theory that solidity is derived from a condition of rest, and a 
kinetic theory of fluids. Independent of Galileo in most respects, 
Descartes expressed the Newtonian First Law quite clearly: by 
itself a body continues in its state of rest or of uniform motion 
in a straight line; curvilinear motion is constrained. We know 
that Galileo also stated this law. Descartes went further than 
Galileo, however, for he considered that rotational motion should 
generate an outward, centrifugal force. In his vortex theory this 
centrifugal force was balanced by the pressure of neighbouring 
vortices, also tending to expand. Each vortex had, in feet, a cen- 
tre which was comparatively empty of matter and consequently 
matter leaving the ecliptic of one vortex was drawn in at the 
poles of a neighbouring vortex. Descartes thought to explain 
magnetism and sunspots by this means. Two other phenomena 
arose in consequence of the vortices : the centripetal attraction 
on matter resulting from the emptiness of the central part of 
a vortex produced the illusion of gravity, and the centrifugal 
pressure of particles of the subtle matter was the origin of light. 
Descartes never really answered the criticism that gravity acts 
along the radius and not along lines perpendicular to the plane 
of rotation and there were other difficulties for his view. In the 
first place he did not consider that the rotating subtle matter 
penetrated the earth. Yet the weight of an object may be shown 
not to vanish beneath the earth's crust. Also Descartes did not 
consider weight as proportional to the quantity of matter and he 
supposed that liquids, through the greater internal motion of 
their panicles, must be inherently lighter than solids. Huygens 
saw these weaknesses. Nevertheless, he followed Descartes on the 
subject of gravity to a greater extent than in other topics. He 
even calculated how fast the subtle matter must move in its 
vortex to produce the known effects. There are other parallels, 
too, in Huygens's theory of the transmission of light. Neverthe- 
less, in the pages which follow it will be seen that Huygens 



remained outside the main stream of Cartesian thought which is 
to be traced in the writings of Rnier, Regius, Rohault, Regis, 
Fontenelle and Malebranche. On the other hand, he was not in 
spirit so deeply opposed to Cartesianism as Leibnitz, whose view 
of the Principia Philosophise was that it was " un beau roman 
de Physique ". 

In this work Descartes supposed that motion could only be 
transferred from one body to another by direct or indirect im- 
pact. His assumption that the total amount of what he called 
the " quantity of motion " in the world is constant was not un- 
reasonable : it conceals the germ of the idea of energy which 
may be traced in the writings of Stevin, Da Vinci and perhaps 
in those of Jordanus Nemorarius. But Descartes made a distinc- 
tion between the speed of a body and its " determination " or 
direction which led him to treat these as separate entities. In his 
third law of motion Descartes stated that if a moving body, tra- 
velling in a straight line, meets another body with less " force " 
to continue its movement than the second one possesses to resist 
it, it will lose its determination without losing its motion. If it 
has more " force " it moves the second body and loses as much of 
its motion as the second one absorbs. Descartes's seven rules of 
impact were all corollaries of this law and so important were 
these in his system that Mouy has remarked: " Les regies du 
choc sont bien les regies du mouvement et la m^canique carte- 
sienne est une theorie de la percussion." It was unfortunate, 
therefore, that all the rules were wrong when compared with 
experimental results. Descartes appears to have known this and 
to have replied in effect: the rules are true for perfectly hard 
bodies moving in uacuo. In experimental conditions the bodies 
are not perfectly hard and move in a fluid. The parts of this 
fluid " corrupt " more or less the bodies immersed in it owing 
to the motion of the fluid particles against the particles of the 

During the year 1652, when he was only twenty-three, Huy- 
gens became convinced of the errors of Descartes's treatment of 
motion and impact. Correspondence with Schooten, Gutschoven 
and Slusius in the period of 1652-7 shows that he was occupied 
in substituting new principles for those referred to above. The 
fact that Huygens's work 1 was not published as a whole in his 
lifetime is certainly not to be attributed to his reverence for Des- 
1 It was entitled De Motu ex Percussione. 



cartes but to his own difficulties over further dynamical prob- 
lems. Actually the work was completed before 1656, for in that 
year Huygens undertook the recasting of his material in classical 
form. Some twelve years, therefore, elapsed between Huygens 's 
completion of the treatise and the communication of his results 
to contemporary men of science (see p. 64). 

Huygens at first limited himself to the study of direct col- 
lision between two equal bodies. He began with two hypotheses. 
The first was Galileo's principle of inertia, also stated by Des- 
cartes; the second stated a principle of symmetry. The latter 
was that if two exactly similar hard bodies collide with equal 
speeds in the same straight line they are reflected back with their 
speeds unchanged. He uttered a warning that this was true in 
respect of the system of co-ordinates used in specifying the posi- 
tion of the bodies at any instant and that the system might be in 
a state of uniform motion. Velocities relative to the system of 
co-ordinates or reference frame he saw must be treated as vectors 
as forces were already treated. His problems were of course 
concerned with momenta, but this fact is a little obscured by his 
geometrical method of treatment in which the velocities alone 




FIG. 2 

are represented. The bodies were supposed to be smooth spheres 
of a perfectly elastic material. They were imagined as suspended 
by vertical strings held by a man standing in a boat. The 
boat or system of co-ordinates could be given a uniform 
velocity to the right or to the left so that the velocities of the 
spheres, as judged by a stationary observer on the bank, were 
ijn effect transposed to a different system or reference frame. 

Thus, if the spheres have velocities v and -v (Fig. 2) and the 
reference frame moves with a velocity -v the first sphere is made 


to appear stationary and the second to approach with a velocity 
of -2V. After impact, with respect to the frame, the velocities arc 
-v and v, but with respect to the stationary observer they appear 
to be -av and o. Thus, when one body is stationary and is struck 
by a moving body of equal mass, the momentum is transferred. 
In the case where the body A has a velocity v l and JB has a velo- 
city -v a where v x > v 2 , Huygens supposed the reference frame to 

be given a velocity - so that, to the stationary observer, the 

velocities appear to be equal and opposite. It followed that the 
real velocities are interchanged. 

To progress in problems where the masses are unequal Huy- 
gens found it necessary to assume as self-evident that when a 
larger body meets a smaller one which is at rest, the smaller 
one is moved and the larger one loses some of its motion. Here 
he was engaged in clearing away some of Descartes's errors, some 
of which arose through his failure to treat velocity consistently 
as a vector quantity. In March 1669, in the Journal des Savants, 
Huygens gave a correct statement of the law of conservation of 
momentum in the form: " There always remains the same 
quantity of motion towards the same side after the quantity of 
contrary motion has been subtracted." Another form of 
law is given in Rule 7 of the DeMotu: "... the common 
centre of gravity of the bodies advances always equally towards 
the same side in a straight line before and after impact." 
That is, the total momentum along the line of centres 
is unchanged. It followed directly from this that the relative 
speeds of approach and separation are equal. 

Much the most interesting part of the De Motu is found in the 
proposition on the collision of bodies when the velocities are 
inversely proportional to the masses. 1 " When two bodies, whose 
speeds are inversely proportional to their masses, meet from 
opposite sides, each rebounds with the same speed with which 
it approached." 

Here Huygens employed the important principle, referred 
to in Part I, that the centre of gravity of a system of bodies 
cannot ascend through any motion of the bodies under gravity. 
His proof consisted in showing that if the resultant speeds were 
other than equal and opposite to the original speeds, the centre 

1 Proposition 8. Huygens did not make a distinction between mass and 
weight in this work. The first use of a distinct term for mass comes in his 
work on centrifugal force (see p. 117). 


of gravity of the bodies could be raised above its original height. 
This, following an idea found in the writings of Stevin and 
Torricelli, he assumed to be impossible, since it involved a 
possibility of perpetuum mobile. In the figure the mass m A pos- 
sesses the velocity AC, while the mass W B possesses the velocity 
BC. These are related: 

tw A = BC 

m B AC 

FIG. 3 

After impact, according to the theorem, the velocity of m A will 
be represented by CA and that of ra B by CB. Using a reductio 
id absurdum, Huygens first assumed that the acquired velocities 
ivere CD and CE, the only necessary restriction here being that 
AC + BC = CD + CE (ignoring signs). Supposing that the 
initial velocities were acquired by vertical descent under gravity, 
Huygens took the required heights to be HA and KB respec- 
tively. These were related through Galileo's equation v* = 2gs 

[Fig- 3)- 

HA _ AC' 

1 : ~KB ~ C& 

Starting with the velocities CD and CE the heights attained 
would be AL and BM such that 

AL _ CD 3 

~HA ~ AC*' 

BM _ CE* 
and KB ~ OP' 



When first elevated ra A and w B have their common centre of 
gravity at N. After being brought to the positions L and M , the 
centre of gravity is moved to O for 

m^.AC = ra B .CB 

The remainder of the proof then consists of a simple demonstra- 
tion that O is higher than N not only for this case but also for 
the case when ra A rebounds with more speed than before and for 
the possibility that it is brought to rest. 

It will be noticed that the idea of converting a horizontal 
velocity into an ascent involves first the relation v* = 2 gh and 
second the idea of energy conserved, i.e., mv 2 = 2 mgh. No 
doubt it was the use of this relation which enabled Huygens to 
solve problems on the ballistic pendulum when in England in 
1661. It marks a distinct advance on Descartes 's approach to the 
law of conservation of energy, since the latter had no use for any 
form of potential energy. This piece of work is accordingly of 
some importance in the history of the concept of energy. 

After this point Huygens was able to give the solution of the 
most general case of impact between perfectly elastic bodies mov- 
ing along the line of centres. The masses being in general 
unequal, Huygens employed his original device and gave the 
reference frame a velocity such that the velocities of the bodies 
bore an apparent relation inversely proportional to the masses. 
His ^geometrical method prevented him from expressing the 
result in a generalized form. The most general law embodied 
in the work on impact is the law of conservation of kinetic energy 
for linear motion in the horizontal plane. This is stated in a 
proposition showing that ra A ^ 2 A -f m B z; 2 B = constant, both be- 
fore and after a collision. The difficulty of dealing with vector 
quantities is overcome by squaring the magnitudes. How Huy- 
gens hit on the quantity mv 2 is not quite clear, but his use of it 
appears to date from about 1652 and he considered this, the 
efficacy or vis viva of a moving body, to be a more fundamental 
quantity than its quantity of motion or, as we should now say, 
its momentum. Leibnitz adopted the same point of view, but 
neither he nor Huygens saw how the vis viva was related to the 
quantity of motion. 

The remainder of Huygens's De Motu Corporum ex Percus* 


siane deals with the effect of placing hard bodies between the 
two extreme bodies in motion so that the impact is transmitted. 
In the simplest case a hard sphere is brought to rest on meeting 
the end sphere in a row and the sphere at the other 
extreme end is set in motion with the velocity originally 
possessed by the first (Fig. 4.) Huygens experimented with glass 
balls suspended like pendulum bobs or rolling in a horizontal 

FIG. 4 

groove. It is well known that he found in the study suggestions 
for a theory of the propagation of light. He showed in the present 
connection that a single interposed mass must be the mean pro- 
portional between the two extreme masses for the velocity 
attained by a third mass to be a maximum. Thus, at first, m x 
has a velocity t> A while V B o and v c o. It may be shown that if 

B = */^*"~C> 

then v c is a maximum for any given values of ra A and ra c . The 
proof is straightforward, although tedious by Huygens's method. 
As has been mentioned, others besides Huygens brought out 
work on the subject of impact. Wallis, in November 1668, read 
a paper to the Royal Society in which he also employed the Car- 
tesian quantity of motion but dealt with inelastic bodies. In 
167 1 he published results for elastic impact, including the formula 

mv m.v, . 

u for inelastic bodies, 

m -f mi 

where v and v l are the initial velocities and u the common 
velocity after collision. In December 1668 Wren also gave some 
empirical rules which resembled some of Huygens's results, 
although arrived at independently. 

For a general theory of impact, clearly systematizing con- 
temporary work, one has to turn to Newton's Principia, where the 
subject is summarized in the introductory Scholium to Book 
I. After the three laws of motion Newton gave a correct state- 
ment of the law of conservation of momentum and stated as 
a corollary Huygens's " law " that the common centre of gravity 
of two or more bodies is unaffected by the actions of the bodies 


among themselves. So slowly did the Newtonian system displace 
the Cartesian, however, that in the English edition of Rohault's 
System of Natural Philosophy, which came out in 1723, Des- 
cartes's original treatment of motion and impact is closely fol- 
lowed and we read, for example, on page 78 : " When a Body 
moves any particular way, the Disposition that it has to move 
that way rather than any other is what we call its Determina- 
tion. 1 ' Gravesande's Mathematical Elements of Natural Philoso- 
sophy (1721) was more up to date and contained a summary of 
Huygens's work on impact without, however, an acknowledge- 
ment of the source. This work was dedicated to Newton and is 
an interesting guide to the scientific heritage of the seventeenth 
century as it was passed on to the eighteenth century reader. 

Centrifugal Farce 

The fact that Huygens early turned his attention to the 
study of circular motion is only another example of the influ- 
ence of Descartes. In his Principia Descartes attempted to 
analyze the motion of a stone placed in a sling. He saw that 
although the stone tended to continue its motion along the tan- 
gent, this was prevented by the tension in the thread. The prob- 
lem was to find the magnitude of this tension and this Descartes 
failed to do. Huygens completed his study of circular motion 
about 1659 and published the more important theorems at the 
end of his Horologium Oscillatorium. The treatise, De Vi Centri- 
fuga, containing the proofs of these theorems and other mat- 
ter, was published posthumously in 1703. By this time others, 
including Keill, Savilian Professor of Astronomy at Oxford, had 
become impatient for the proofs of the theorems and had sup- 
plied many of them. Newton arrived at the fundamental 
formula for the acceleration of a particle describing a circle 
independently of Huygens. 

Huygens began his treatise with a statement of Galileo's con- 
clusions concerning descent under gravity. " Gravity," he said, 
" is the tendency to fall " and the tension in the thread from 
which a heavy body is suspended results from this tendency. A 
Constant acceleration must mean that " the spaces traversed in 
different times by bodies starting from rest are to each other as 
the^quares of the times ". The tension considered in the case of 
the sling is therefore a real 
force, as real as a force which 
produces an acceleration in 
the direction of the existing 
motion. Indeed, the measure 
of the force is supplied by the 
acceleration and, he should 
have added, the mass moved. 
There is then an acceleration 
towards the centre in the case 
of circular motion. FIG. 



In the figure (Fig. 5), Huygens considered the successive 
points reached by a particle liberated at some point of its circular 
path and free to travel along the tangent. 

These, at successive equal intervals of time, would be K, L, N 
in the figure, the distances BK, BL, BN being equal to the 
lengths of the arcs BE, BF, BM respectively. If the intervals of 
time are small, BK, BL, BN approximate to BC, BD, BS respec- 
tively; C, D, S being points on the tangent where this is cut 
by the radii through E, F, and M. EC, FD, and MS are then the 
distances separating the two paths at successive instants. With- 
out explaining his reasoning further Huygens then stated that 
these distances approach as a series, the series of squares i, 4, 9, 
1 6, etc., and thus resemble the successive distances fallen under 
gravity. The argument is somewhat reminiscent of Galileo's 
study of a projectile, which, starting horizontally, under the 
acceleration of gravity describes a parabola (Discourses on Two 
New Sciences, Fourth Day), but he may have got at the result as 
we should now by considering a parallelogram of velocities 1 . As 
a mathematician he was rather more attracted by the problem of 
the nature of the curves EK, FL, and MN, which he saw were the 
evolutes traced out by the movement of a stretched thread 
wrapped along the arc BE, BF, BM, etc., but he reverted to the 
physical aspect of the subject. The distances traversed (EC, FD, 
MS) in successive instants being approximately in the ratio of a 
series of squares, he wrote, " this tendency of which we have 
spoken is absolutely similar to that with which heavy bodies 
suspended by a thread try to descend. Whence we concluded 
also that the centrifugal forces of unequal bodies moved around 
equal circumferences with the same speed are among themselves 
as the weights or solid quantities " inter se sicut mobilium 
gravites, seu quantitates solides. Professor Crew (The Rise of 

a dt 
a.dt = v& 

8' = (ddJ and <o = v/r 

.'. a = v*/r. Under gravity a = v a /25. 


Modern Physics), has remarked that this is probably the earliest 
suggestion of a distinction between mass and weight. 

From this original beginning some simple conclusions follow 
immediately. The centrifugal force varies directly with the 
radius of the circle if the angular velocity and mass are constant 
(Fig. 7), it varies as the square of the tangential velocity at any 
instant (Fig. 8) and inversely as the radius (Fig. 9) : 

(a) For equal angular velocities : 


FIG. 7 

FIG. 8 


FIG. 9 





ije. the centrifugal force varies 
with the radius (prop. i). 

(h v, _ BE 

"vl ~ BF 



and, by his original relation, 

CE __ CB' _ V 
DF = DB T = u?' 

(c) Equal masses have equal linear 
velocities: BD CF - v. 

LetCE = v' 
v^_CE^_ AC 
v "~ 


But F a 

F' : 


(prop, i) 

and F' AC' , 

F = JR (P r P- 2 ) 

BA AC 1 


AC AB' = AB 



There does not seem to be much necessity to quote the other 
propositions on ordinary circular motion since they follow 
simply from the above. There are, however, some interesting 
studies of the conical pendulum : problems which may have been 
a reason for Huygens's taking up the whole subject. As has been 
mentioned earlier he made use of the conical pendulum in some 
of his clocks. 

This section commences with the 
statement of two simple theorems or 
lemmata on the equilibration of smooth 
bodies on inclined planes by means of 
tensions in horizontal threads. Thus in 
Fig. 10 (lemma i) the weight D must be 
to the weight C as the perpendicular 
This relation is not proved; Huygens 
it is one which is well known. It is 


FIG. 10 

RF is to the base FA 
says, justifiably, that 

used immediately in an interesting theorem on the revolution 
of a body on the inside of a paraboloid of revolution; all revolu- 
tions of a body travelling in horizontal circles on the surface of 
the paraboloid are completed in equal times irrespective of 
the amplitude of the circle. This theorem states the theoretical 
conditions to be fulfilled by a conical pendulum used in a clock. 
The point about Huygens's proof of his theorem is that, true to 
his conception of centrifugal force as equivalent to any other 
kind of force, he substituted it for the tension in the thread of 
the above lemma. At any point on the paraboloid, then, the cen- 
trifugal force must be that which will maintain the body 

against the force of gravity along 
the tangent. As he showed: 

F _ HG = HK 
L I mg ~ GF ~ KL 

K / But, by a property of the para- 

bola, KL is constant (Fig. n). 
Thus, for any two positions of a 
body on the paraboloid: 



and thus the times will be equal 
FIG. 1 1 (proposition i). 

In subsequent propositions, Huygens derived all the simple 


relations which might be expected: the various factors of vertical 
height, length, inclination to the vertical and angular velocity 
all being considered separately. Just as he nowhere used the 


formula in this form so he did not give the formula 

cos for the period of the conical pendulum. The work 


is consequently somewhat tedious for the modern reader. Near 
the end Huygens showed that when the angle with the vertical 
made by the thread of a conical pendulum is 2 54' the period is 
equal to the time of vertical fall from a height equal to the 
length of the pendulum. The work is completed by a consider- 
ation of circular motion in a vertical plane. Problems which 
are still common in modern text-books are solved here for the 
first time. Huygens gave also the necessary conditions for 
Galileo's experiment with the intercepted pendulum (Fig. n). 
This was an instructive experiment since it illustrated the 
relation gh = v 2 /2 and drew attention to the fact that the 
particular path makes no difference to the height achieved. This 
is sufficiently explained by the figure. 

FIG. 12 

As has been mentioned, Huygens followed the old Aristo- 
telian view that circular motion is a distinct form. This accorded 
also with Galileo's view, for although he himself discovered 
parabolic motion he did not include it with the other two funda- 
mental motions. It seems as if both Galileo and Huygens 
supposed the elliptical paths of the planets to be reducible to 
simpler components. At any rate Huygens for a long time re- 
served a special place for circular motion, considering it alone to 
be absolute and not merely relative in nature. After the appear- 
ance of Newton's Principle*, however, he came out against the 


idea of any absolute motion and opposed the idea of absolute 
space or time. In 1694, he criticized Leibnitz for holding to the old 
idea of absolute motion, but rejected the suggestion that he owed 
his point of view to Newton's Principia. Mariotte also he con- 
sidered was in error in attempting to distinguish the vitesses of bodies. This, of course, raised the question of how we 
may know when bodies are relatively at rest. Huygens answered 
this in two ways : the bodies must be free to move and yet retain 
their relative positions with respect to each other and to their 
background; alternatively, by connecting the bodies by threads, 
one can dispense with the background, for if rotational motion 
exists, tensions will be set up in the threads. 

The effect of the diurnal rotation on the shape of the earth 
was, of course, a problem which Huygens could not ignore. 
Kepler, in his Epitome Astronomic Copernicx (1609), seems 
first to have proposed that the centrifugal force must be equili- 
brated by an " attractive virtue " and Huygens followed this 
lead in 1666. He calculated how much slower a pendulum clock 
would go in a latitude of 45 when compared with its rate of 
going at the poles. Supposing the earth to be perfectly spherical 
for the purpose of the calculation this amounts to a comparison 
of oscillation in a field of g units as against one g' = g - F cos a 
where F is the centrifugal force acting on unit mass at a latitude 
. Clearly we may put 

where r = R cos a and R is the radius of the earth, T the period 
of the diurnal rotation. Substituting the values for R and T, for 
a latitude of o the centrifugal force is a maximum and works 

out to Q approximately. This agrees with Huygens's 


estimate that the rate of rotation would have to be seventeen 
times as fast for the centrifugal force to equal gravity at the 
equator, but his estimate for the slowing down of the clock does 
not agree with a modern estimate. His attempt to deduce the 
form of the earth was, however, interesting since it laid down 
useful principles for the theory of the equilibrium of fluids. 
Huygens laid down the principle that a mass of fluid is at rest 
only when its surface is at each point perpendicular to the 
resultant force acting at that point. Later surveys carried out in 


the eighteenth century confirmed his general conclusions, but 
not the extent of flattening at the poles which he had calculated. 
For a time Cassini opposed the views taken by both Huygens 
and Newton as to the form of the earth resulting from rotation. 
A cartoon of the period is reproduced in Professor Capri's 
edition 1 of Newton's Principia, in which the rival pictures are 
shown (Fig. 13). 

Newton Cassini 

FlG. 13 

The investigation by which Maupertuis later disproved 
Cassini's conclusions gained for him the title of the "earth- 
flattener ". This was, however, complimentary when compared 
with the other epithets he received from Voltaire. 

An account of the dispute which broke out between the 
Newtonians and Cartesians on the shape of the earth was given 
by d'Alembert in L'Encyclopddie of 1751-65 (Vol. VI). 

1 The Mathematical Principles of Natural Philosophy, a reprint of the English 
translation by Motte (1729) with some additional notes. 


In some early studies on the distribution of forces in threads 
which support a number of masses, Huygens used a fundamental 
principle which was closely related with the one he used so much 
in Dynamics. This stated that, for a system of bodies connected 
by threads and in equilibrium, a very small displacement (com- 
patible with the connections) cannot cause the elevation of the 
centre of gravity of the system. This idea, a variant of the idea 
of virtual work, had been used by Torricelli and Pascal, but 
Huygens made it the basis of a general method. He used it, for 
example, in an independent proof of Jordanus's theorem on the 
inclined plane (see below), and it was, of course, a form of this 
principle which was employed so successfully in his study of the 
centre of oscillation. To some extent Huygens followed the lead 
of Galileo's Discourses but Stevin's Beghinselen der Wieghconst 
(1586) and Hypomncmata Mathematica (1608) were also im- 
portant and supplied him with some of his early ideas. It is well 
known that Stevin regarded the impossibility of perpetual 
motion to be a principle on which a treatment of equilibrium 
under gravity could be based. As will be seen, Huygens's work 
carried this idea further and in effect introduced the idea of the 
conservation of energy in mechanical systems (see p. 154). 

In the Discourses on Two New Sciences (Second Day), 
Galileo had given a "proof" of Archimedes' theorem on the 
simple lever and had considered also the conditions under which 
a uniform beam is fractured by bending. Both of these problems 
were considered by Huygens in some early studies dating from 
about 1662. The defect of Archimedes' work on the lever was 
that in effect he had to assume the very law he set out to 
" prove ". Stevin, and later Galileo, shortened the " proof " but 
neither, in Mach's opinion 1 , escaped the original difficulty; both 
employed " the doctrine of the centre of gravity in its most 
general form, which is itself nothing else than the doctrine of the 
lever in its most general form ". Huygens did the same. His 
exercise on the subject is of interest since he employed a device 

1 . Mach. The Science of Mechanics (trans McCormack, 1907). 


which was virtually the taking of moments about an axis which 
was chosen arbitrarily. Wallis, alone with Huygens, shares the 
honour of introducing the idea of a moment into mechanics. In a 
letter to Huygens dated January i, 1659, Wallis spoke of the 
momenta of the elements of a surface about a certain axis as 
forming a series in which each term was a weight multiplied by a 
distance. He used the idea in finding the centre of gravity of 
solids of various forms. As has been mentioned, Huygens met 
Wallis in London in 1661 (see p. 45), and it seems very probable 
that he was indebted to him for the germ of some of his ideas. 
Much the same sort of problem as the determination of centres 
of gravity was met by Huygens in evaluating the quantity Hmr 
for the compound pendulum. Such important ideas were, then, in 
use long before they became pan of the published work on 
Mechanics. Varignon's Projet d'une Nouvelle Mecanique (1687) 
was actually the first book to contain the modern idea of 

In some manuscript notes on the fractures of beams Huygens 
showed an approach to the modern idea of work done. Under 
gravity the centre of gravity of a system will descend as far as 
possible, that is, the work done will be a maximum. Huygens 
used the term descensus gfauitdtis in this connection. When the 
system is in equilibrium the first principle, stating that the 
centre of gravity does not rise, is to be applied. 

The problem! of demonstrating the equilibrium of two 
smooth weights on inclined planes and connected by a thread for 
the conditions 

m l __ AB 

^T~ """ Tar 
m-j *- yo 

was solved by the writer of 
the work Jordani Opusculum A 
dePonderositate-z thirteenth FlG - "4 

century work and it was given an elegant solution by Stevin, 
who simply applied the principle that perpetual motion is im- 
possible. The independent solution given by Huygens is therefore 
not referred to in modern works. Nevertheless, it shows clearly 
the usefulness of his fundamental axiom that the centre of 
gravity of a system of weights does not rise through any motion 
of the weights under gravity. 


In the figure (Fig. 14) m l and m a are related as shown above 
and are considered to undergo a small displacement. It is required 
to show that the centre of gravity G is not disturbed. The point 
G in the horizontal line DE must of course be such that 




BE ~ 




BE ~ 


BL ~ 


Taking DD' = EE' and D'L parallel with DE we have 



an BE = BL = EL = ~EL 


EE' E'G 

EL ~ D'G 

so that G is also the centre of gravity of m l and m, in their new 

In the history of statics, Wallis, by publishing his Mechanica 
sive de Motu in 1669, with successive parts in 1670 and 1671, con- 
tributed a systematic treatment which must be ranked above 
Huygens's work in importance. Huygens, as has been noted, 
was more interested in dynamical problems. The work of these 
two men shows, however, some interesting resemblances. Both 
employed the principle of virtual displacements and extended 
the idea of force to all kinds of forces without distinction. Up 
to this time gravitational force alone had been considered. 

The Treatise on the Pendulum Clock: 

The Horologium Oscillatorium. Part One 

Construction and Use of the Pendulum Clock 

The great Danish astronomer, Tycho Brahe, used clocks in 
astronomical observations about 1580, but he was perhaps not 
the first to introduce time measurement into observations in 
Astronomy. Certain early Arabian astronomers, according to 
Robison, used the simple pendulum for measuring short inter- 
vals, but the evidence for his supposition is not very clear. No 
good means really existed until the pendulum clock was in- 
vented, and by the middle of the seventeenth century the need 
of time measurement had become urgent. Galileo used a water 
clock or clepsydra in his experiments on natural acceleration. 
He clearly knew of the property of the simple pendulum : that its 
period is (very nearly) independent of the amplitude of swing. 
Indeed, it is very probable that this was known to Copernicus 
(see p. 99). The story that Galileo discovered the property of 
the simple pendulum from observation of a lamp swinging in 
Pisa cathedral is not very well established, but the belief is un- 
doubtedly kept alive by a passage in his Discourses on Two New 
Sciences, where he said that he had often observed such oscilla- 
tions in churches. 

Tycho Brahe's clock was a crude instrument employing a 
balance or verge escapement. Such a regulator has no natural 
period, but is adjusted empirically by altering the moment of 
inertia. Although such clocks were used in public buildings, 
perhaps from the i3th century, they were ill-adapted for the 
observatory. Tycho found it necessary to correct for the tempera- 
mental qualities of his instruments at Uraniborg. It is accord- 
ingly very interesting to find that that universal genius Leonardo 
da Vinci, before the end of the fifteenth century, sketched, even 
if he did not construct, an escapement employing a pendulum. By 
the time of Galileo it seems probable that the idea of applying 
the pendulum was present in many minds. The design attributed 
to Galileo shows considerable mechanical ingenuity and owes 
less to the old balance clock than did da Vinci's sketch. 




J. Drummond Robertson, in his excellent book, The Evolution of 
Clockwork (1931), compared it with Huygens's design to the 
latter's disadvantage. Galileo's mechanism, he wrote, " is far 
more subtle and ingenious than Huygens's method of control- 
ling the regular action of the old faulty escapement by the 
attachment of a pendulum in place of the balance ". Neverthe- 
less, Huygens's theoretical study of the problem was far more 
profound and indeed left little to be added, and his escapement 
also seems in practice to be much superior to that of Galileo. 
His practical contributions were superseded by the invention of 
the anchor escapement and later by the dead beat escapement. 
The latter gives an impulse to the pendulum near its zero 
position and exerts little frictional drag over the rest of the 
swing. The ideal of the free-swinging pendulum was not 
attained until the end of the nineteenth century. 

Huygens at first connected the pendulum rod DF (Fig. 15) 
to a circular balance by means of a two pronged fork or crutch. 
The part DE was flexible. This arrangement was, however, 
soon abandoned and the clock described in the Horologium of 
1658 showed some improvement, as a comparison of the figures 
(Fig. 15 and 16) will make clear. The circular balance has gone 


Fie. 15 

Fie. 16 



and the pendulum operates the verge with pallets V by means of 
the fork Q and the wheel P engaging the pinion O. The crown 
wheel is vertical. A vertical crown wheel was, however, not in- 
variably used. Samuel Coster, clockmaker to Huygens at the 
Hague, made some clocks with a horizontal crown wheel and one 
of these, described by J. Drummond Robertson (loc. cit.), was 
driven by a spring and not by weights. And although the clock 
bears the date 1657, it bears metal plates or " cheeks designed to 
correct the period for large swings of the pendulum. These 
plates may, of course, have been added subsequently. As 
J. Drummond Robertson points out, it must be supposed that 
Huygens began with a horizontal crown wheel but changed this 
to the vertical position in his Horologium so as to interpose the 
wheels O and P in the figure. As for the use of curved plates on 
either side of the pendulum it is clear from Huygens's corre- 
spondence that he was uncertain whether to design die clock so as 
to employ a pendulum oscillating over a small arc or to employ 
the curved "cheeks". In the Horologium the cheeks are 
omitted. Huygens explained to Petit that he could not, 
therefore, dispense with the wheels O and P as the latter 
suggested since these had the effect of restricting the arc of the 
pendulum oscillations. 

There is an interesting letter from Huygens to Boulliau 
written in December 1657, in which he enquired about clocks 
which the Grand Duke Fernando de Medici was reported to 
have under construction " in order that I may know if they also 
use a pendulum. A year ago yesterday exactly, I made the first 
model of this kind of clock and in the month of June [1657] I 
began to show the construction of it to all who asked me for it 
and among whom perhaps was someone who sent news of it to 
Italy ... In a very few days we shall see a much larger form of 
the clock in a belfry of the village which is near the sea half a 
league from here [the village of Scheveningue] . The pendulum 
will be a i feet long and will weigh about 40 or 50 pounds . . . /' 
The Medici Palace clock was for some time put forward as being 
of earlier date than Huygens's clocks. Modern writers do not 
accept it as antedating Huygens's invention, and it may in fact 
have been a copy. Against this is the fact that the pendulum of 
the Medici clock was rigidly attached to the escapement a 
feature of both da Vinci's and Galileo's design. This was a 
defect, since if the pendulum was light the oscillations could 



easily become forced and the mechanism would then resemble 
an inferior balance c|pck. "... the great merits of his inven- 
tion " wrote J. Drummond Robertson of Huygens's clock/' were 
the free suspension of the pendulum by means of a cord or a 
steel spring, with the crutch as the means of communication 
between the pendulum rod and the verge; contrivances which 
have continued in use ever since in all clocks with the anchor 

FIG. 1 8 

FIG. 19 


When one turns to the Horologium Oscillatorium of 1673, 
one finds that Huygens gave up the two toothed wheels inter- 
posed between the pendulum and the verge with pallets and 
reverted to metal cheeks to render the oscillations truly 
isochronous for all arcs. In 1659 he discovered the theoretical 
form for these cheeks and showed that it was a cycloid arc. It 
became necessary to revert to the first design employed by Coster 
in 1657, and to go back to the horizontal crown wheel as shown 
in the figure. Figure 17, reproduced from the Horologium 
Oscillatorium, shows the flexible suspension between the cheeks 
more clearly. The second figure (Fig. 1 8) shows a pendulum clock 
with a tapered cylindrical pendulum and the method of using a 
" maintaining weight " as invented by Huygens. This latter 
device made it possible to rewind the clock without stopping it. 
The cord applying the weight passed over a spiked pulley D in 
the large figure. The following quotations are taken from the 
Horologium Oscillatorium. 

The opposite figure [Fig. 19] represents the clock seen from 
the side, snowing first two plates AA, BB, half a foot long or a 
little more and twenty-two inches wide, whose corners are joined by 
four little columns so that the plates are distant from one another 
by one-and-a-half inches. In these plates are placed the axes of 
the principal wheels on both sides. The first and lowest wheel is 
that which is marked C, incised with 80 teeth, and to the axis of 
which is fixed the pulley D spiked with iron points to hold the 
cord with the weights attached, the reason for which is given 
later. The wheel C thus turns by the force of a weight; this moves 
the nearest pinion E which has eight teeth and at the same time the 
wheel F which has 48 teeth and is attached to the same axis. The 
latter moves another pinion G and a wheel H on the same axis, 
the numbers of whose teeth are the same as those of the 
preceding pinion and wheel. But this wheel is of the kind that our 
artisans call a crown wheel. By its teeth are turned the pinion / 
and at the same time the wheel K which is on the same perpen- 
dicular axis. This drum has 24 teeth and the wheel 15 which 
are made like those of a saw. Above the centre of the wheel K is 
placed horizontally the rod with pallets LM whose ends are sup- 
ported on each side by the plates N and P separately attached to 
the plate BB. In the plate NP the part Q should be noticed, pro- 
jecting towards the base, through an oblong opening in which 
the axis LM passes and which besides keeps in the vertical position 
the axis which we have said is common to the wheel K and the 
pinion /. In the plate BB a large opening is made through which 
can pass the end of the rod with pallets LM, which, inserted by 
its pointed end into the plate P moves thus more freely than if it 


were supported by the plate BB itself and were at the same 
time prolonged through it; an extension is necessary so that the 
crank S may be attached to it so as to oscillate with it. For this 
is an oscillatory or reciprocating movement since the teeth of the 
wheel K make contact in turns with the pallets LL in the 
customary way and which needs no further explanation. 

As for the crank S, the lower pan of which is bent back and 
pierced with an oblong hole, this engages the iron rod of the 
pendulum to which the bob X is attached. This rod is suspended 
from above by a double thread between twin plates of which only 
one, T, is visible here. For this reason we have shown at the side 
a second figure intended to make dear the form of each and the 
general manner in which the pendulum is suspended. It will be 
necessary, however, to return to this subject and consider the true 
curve of the plates. 

We will now turn to the movement of the clock for we shall 
explain the other parts of the figure later. It is easily seen that the 
pendulum VX, when once it has been set in motion by hand, 
maintains its motion through the force of the wheels driven by a 
weight; and at the same time the fixed period of the pendulum 
prescribes for all the wheels and consequently the whole clock, 
the law and pattern of the movement. Indeed the crank, however 
slight may be the movement communicated to it by the wheels, 
not only follows the pendulum which moves it but also contri- 
butes to that movement a short impulse at each come and go. It 
maintains then a movement which without this assistance would 
decrease little by little, in part at least owing to the resistance of 
the air, and come to rest. Indeed, the pendulum having the 
property of always following the same course, and not being 
diverted from it unless its length change, the wheel K is not per- 
mittedonce we have obtained the equality of which we spoke 
above by means of the curved plates between which the pendulum 
is suspended to go now more quickly and now more slowly 
although in common clocks it often endeavours to do so; here its 
teeth must necessarily pass one after another in equal times. It is 
dear that the revolutions of the preceding wheels, like those of the 
hands which come last, are also rendered uniform seeing that all 
the parts move proportionally. Consequently if there is any fault 
in the construction or, on account of a change in the timing, the 
axes of the wheels turn with more difficulty, so long as this 
difficulty is not enough to cause the complete stopping of the 
dock there will be no reason to fear any inequality or retardation 
of the movement; the clock will always measure the time correctly 
or not measure it at all. 

After completing his description of the mechanism, Huygens 
gave directions for adjusting the clock so that it completes 
twenty-four hours in the mean solar day. This is the ordinary 



adjustment of a clock for everyday use. For use in the obser- 
vatory the sidereal day and not the mean solar day is the unit 
required. There is about four minutes difference between these 
days but the correction is not constant since the solar day is not 
constant. Huygens gave a table showing the individual correc- 
tions for comparing a given (solar) day with the sidereal day. 

In astronomy the measurement of the time of passage of a 
star across the meridian can replace the measurement of meri- 
dian altitude which is more difficult and, in the seventeenth cen- 
tury, the latter was always rendered somewhat uncertain through 
the absence of reliable corrections for atmospheric refraction. 
The history of the idea goes 
back some way. To explain it, 
it is necessary to refer to the 
diagram (Fig. 20) which shows 
the planes of the equator and 
the ecliptic intersecting the 
celestial sphere. As is well 
known, the position of a star 
in the sky is not usually re- 
corded by its altitude and 
azimuth because these, unlike 
its right ascension and declin- 
ation~or its celestial longitude 
and latitude vary on the Earth's surface. In the figure O is the 
position of an observer and the celestial equator and the ecliptic 
intersect in the equinoctial points y and - r ^. The plane of the 
ecliptic is of course the apparent path of the sun among the 
stars and the equator is inclined to this at the supplement of the 
angle of inclination of the Earth's axis to the plane of its orbit, 
viz., 32 27'. If 5 is a star we may record its position by stating 
its right ascension (arc ^M) and declination (arc MS) or its 
longitude (arc ^Q) and latitude (arc QS). All these arcs are 
measured in angular units except the right ascension, which is 
more often measured in units of time. For since the celestial 
sphere appears to revolve once in twenty-four hours, any given 
star will move 15 in one hour. The interval between the transit 
of the vernal equinox (the First Point of Aries) and that of a 
star across the meridian therefore gives its Right Ascension in 
hours. For this reason the great circles passing through the poles 
P and P A and drawn at 15 intervals from the equinoxes are called 


Hour Circles. Right ascensions may thus be stated either in 
hours, minutes and seconds, or in degrees. 

It is obvious that when recording the observed position of a 
star by its altitude and azimuth preliminary to reducing these 
to one or other of the spherical co-ordinates, the exact time at 
which the observation was made must be noted. This was at first 
done by recording the altitude of the sun or some other standard 
star. Purbach and Regiomontanus used this method in the 
fifteenth century. Bernhard Walther, a pupil of Regiomontanus, 
is sometimes said to have been the first to use a clock driven by 
a weight for scientific purposes. His clock was useless for inter- 
vals other than about an hour, however, and the first consistent 
use of time intervals was due to Tycho Brahe. Tycho used quad- 
rants to observe altitudes and usually found the distance along 
the equator from the meridian with an armillary sphere. His use 
of a meridian quadrant to observe transits required the measure- 
ment of the time interval between the transit of the star and that 
of the equinoctial line. 

In 1667 Huygens described a method which is similar to the 
well-known method of " equal altitudes " for finding the time 
of a meridian passage of a star. Roemer in 1690 really established 
the method of obtaining right ascensions and declinations by the 
use of transit telescopes. He also set up an instrument having 
altitude and azimuth circles for the observation of stars at equal 
altitudes on both sides of the meridian. The clock he used bears 
somfe resemblance to that of Huygens. One may take it for cer- 
tain that the close association of both men with the Paris observ- 
atory led Roemer to use a clock of Huygens's design. Until the 
invention of the anchor escapement (1680) Huygens's clocks were 
far the most accurate available and in Paris they probably con- 
tinued to be preferred for many years. 

While discussing this subject of time measurement in astro- 
nomy one can see how the clock promised (and later supplied) a 
solution to the problem of determining longitude at sea. On 
March 2 1 a sidereal clock on the meridian of Greenwich agrees 
with a mean time clock. For a place not on this meridian the 
sidereal time must be corrected by the addition or subtraction of 
9.8565 seconds for every hour of longitude difference according 
as the place is west or east of Greenwich. Huygens's own direc- 
tions for the use of the marine clock at sea ignored the slight 
difference between sidereal and mean solar time. These direc- 



tions were that the clock should be set going by mean solar time 
at the starting point of a voyage and that the solar time at 
sea (from the altitude of the sun) should be compared with the 
standard time kept by the clock. To the latter it was of course 
necessary to add the correction for the solar day required to take 
into account the inequality of the solar days. If, then, the mean 
solar time given by the clock after correction is earlier than the 
observed time, the ship has moved east, if it is later the ship has 
moved west. Each hour difference is equivalent to 15 of longi- 
tude. Figs. 2 1 and 2-2 show certain features of one of Huygens's 
later marine clocks. 

FIG. 21 

FIG. 22 


The Horologium Oscillatonum. Part Two 
Oscillation in a Cycloidal Arc 

This part of the Horologium Oscillatorium contained the first 
thorough treatment of oscillatory motion ever given. The study 
of accelerated motion in curved lines was begun by Galileo, and 
it is worth while looking at his work, so neatly was it completed 
by Huygens. Both of these writers treated their problems as a 
branch of " geometry ". It was only after the principles of New- 
ton's Principia had been absorbed that mathematicians 
developed mechanics on the concepts of mass, force and impulse. 
The difference is perhaps more apparent than real, for Galileo 
implicitly and Huygens explicitly recognized the quantity 
termed mass as distinct from weight and concentrated on velocity 
and acceleration, both capable of geometrical representation. 
Huygens really developed a treatment of mechanics on the basis 
of work done what we now term the energy equation but this 
was never fully appreciated. 

In the Discourses on Two New Sciences (Third Day), Galileo 
gave his classic propositions on naturally accelerated motion. He 
began with the assumption that the law of acceleration is simple 
and derived the well-known equations on fall from rest under 
gravity. He showed, among other things, that the times of 
descent of a smooth body over inclined planes of the same vertical 
height are proportional to the lengths of the planes : also that 
chords of the same circle are the paths over which descent takes 
the same time. The latter theorem was employed by Huygens 
and Galileo's proof of it may well be given here in symbolic 

In Figure 23, B and C are any two points on the circumference 
of a circle, centre O and radius r. AF is the vertical diameter, 
AD and AE the vertical heights of AB and AC, and AI is 
the mean proportional of AD and AE. Putting AD~x lt 
AE SB # 3 , AI = ra> we have, therefore, m = >/*i* 3 . 



Now AC' = x* + CE 1 A 

and CE? = CE.EC' = r 3 - OE* < 



= AT a 2 + r* - 

/4B a 

n x~ m 

But = . 


FIG. 23 

For the times of descent along the inclined linfcs we have the 
equations _ 2 

where ^ and a 2 are the accelerations produced. 




AC. sin Q! 
t* AB. sin a 2 



" AB 2 



and by (i) above = i. 

Galileo actually proved the relation (ii) in a separate proposi- 

Galileo pointed out that the speed acquired in descent over 
an inclined plane is always such as would enable a body to re- 
ascend another plane of equal vertical height. His experiment 
with the intercepted pendulum 
illustrated this point (see p. 121). 
He went astray, however, in sup- 
posing that a circular arc is the 
path of quickest descent under 

It is usually supposed that 
Galileo considered the simple 
pendulum made isochronous 
oscillations in all arcs and that 


the period was quite independent of the amplitude. This is cer- 
tainly assumed in the Discourses, where Galileo comments on the 
fact that descent from A to B over the inclined plane (Fig. 24) takes 
longer than descent along a circular arc CB. He then added: 
" As to the times of vibration of bodies suspended by threads 
of different lengths, they bear to each other the same proportion 
as the square roots of the lengths of the threads . . . ". There 
is no record in his writings to show that he observed the dis- 
crepancy of the periods of large and small swings. The Floren- 
tine Academicians, however (Essayes of Natural Experiments 
made in the Accademia del Cimento, translated by R. Waller, 1684), 
said that Galileo observed the " very near equality " of the 
swings. Mersenne, in his work Les Nouvelles Pensees de Galileo 
touchant Les Mecaniques et la Physique, said that Galileo did 
not observe the discrepancy. " If the author had been more exact 
in his trials," he wrote, he would have noticed it. His regrets ap- 
pear somewhat out of place, however, when one considers the 
fertile ideas Galileo drew from his study of the pendulum; an 
exact relation between the amplitude and the period could have 
been only a hindrance at this stage. Enough is known of Gali- 
leo's methods for us to be sure that an element of idealization 
came into his treatment of the pendulum, so that he perhaps 
intentionally ignored small experimental deviations from the for- 
mulated law. The laws of nature, he believed, must be charac- 
terized by their simplicity. 

Huygens did not study the simple pendulum exhaustively. 
After showing that the oscillations could be regarded as isochro- 
nous only when the arc was small he turned his attention to the 
problem of the tautochrone, that is, the curve over which all 
oscillations take the same time under gravity. He showed that 
the cycloid satisfies these requirements, being the curve traced by 
a point on the rim of a wheel which rolls along a horizontal sur- 
face. It was left to Lagrange and Laplace to complete the study 
of the simple pendulum and derive some form of relation be- 
tween the period and the amplitude. The former's Mtcanique 
Analytique (Chap. 2) (1788) and the latter's Mecanique Celeste 
(Chap. 2) ( 1 799) should be consulted. 

Throughout this part of the Horologium Oscillatorium Huy- 
gens's debt to Galileo is very clear. The first nine propositions, 
in fact, are really a clear resume of Galileo's work on natural 
acceleration with some small additions. A good deal of ground 



had to be cleared before Huygens could get to grips with the 
central problem. Some geometrical properties of the cycloid, a 
method of drawing a tangent to the curve, and the setting of 
limits to the length of an arc of a circle occupy the next few 

FIG. 25 

pages. Then in Proposition 22 Huygens showed quite simply 
that if two cycloidal arcs of equal vertical height are considered, 
descent from rest is quicker over the steeper curve. In Fig. 25, 
BD and EF are the arcs, equal in vertical height h but one less 
steep than the other. 

By a property of the cycloid the tangent at any point L is 
parallel with the chord NA of the generating circle. L and M 
are taken such that L is the same vertical height below B as M is 
below E. Transferring the arc EF to the position ef the point M 
is at m, level with L. Since the inclination of the tangent at ra is 
given by the chord OA and for all points such as L and m the 
inclinations of the tangents to the steeper curve are themselves 
steeper, the time of descent must be shorter over BD than over ef. 

In the next proposition Huygens compared the time of 


FIG, 26 


descent over a short segment of a tangent to the cycloid with that 
over an inclined plane of equal height. Supposing a body to be 
released at B in Fig. 26, and to descend along the arc BG, he com- 
pared the time of descent over MN with the speed attained at G 
with the time of descent over OP with the mean velocity acquired 
during descent over the distance BI, the line BI being the tan- 
gent to the cycloid at B. Using Galileo's relation v 2 = igh for 
vertical descent, Huygens took the velocity of a body after des- 
cent over BI to be \/2g. ir or 2 fgr and the velocity at G to be 
One half of the former is thus 



By similar triangles 

FH ~ FE 
FA _ FH* 

and ~ FF 


_ _ 

~FH ~ FE ~ 

l^ FX 

Hence = jg- 

t t MN ^ Vl FX MN 

Now _ = ^ . ^_ = __ . _. 

The remainder of the proof consists of showing by geometry 



OP " HE ' 
t, FX HX ST 

t t ~ HE ~ HE ~~ QR 
The last equality follows since HE = HX cos 6 

QR = STcos*. 

It appears a somewhat curious relation to set out for but the 
time of descent over a cycloid arc is later to be obtained by 
reference to a relation between the time of descent over a series of 
tangents ST, and the time of fall along an intercept on the axis 
FA. Unfortunately, the following theorem is unsuitable for re- 
production here in its original form. Anyone who looks up the 
work in the original will see that here is an interesting method 
of exhaustion pushed to its limits. 


In Fig. 27 a large number of tangents are drawn to the cycloid 
and to the circle. It is required to show that the time of descent 
t l9 along an arc BE, is to the time of descent f a , along the tangent 
BI (with a uniform speed equal to the mean speed acquired over 
Be) as the arc QH is to QG. The intercepts made in BI being sup- 
posed equal, the time interval S/ 2 for motion along each is a fixed 
quantity. The relation with the previous theorem is then best 
shown by taking t l to be the sum of the intervals 8*1, required for 
descent over the tangential elements M^N^ M 2 N 3t etc. 

FIG. 27 
The equation of the previous theorem is then written in the 

81, = S lTl 

dt* QR 

where QR is the interval between the parallels measured on the 
diameter QA. Dividing this equation by the number of elemen- 
tary tangents, n, we get 

and there are n of these relations altogether. Adding these 
equations we get 


n^ a = QG~' 

In the limit, when the elementary tangents are very small and 
very numerous, 257 = QH, 23^ = ^ and we have 

!i . OIL 

* "" QG- 

Huygens was then able to complete his demonstration by 



showing that the time of descent from any point on a cycloid 
to the lowest point has, to the time of descent along the axis, 
the ratio of the semicircumference of a circle to its diameter. 

FIG. 28 

Applying the previous theorem to the case of descent to the 
lowest point 

^ = ar ^^ 
t M 

where t M represents the time of descent with the mean speed 
acquired over the tangent BG. This, however, is the same as the 
time for naturally accelerated fall along BG. Now BG is equal 
and parallel with EA (by a property of the cycloid) (Fig. 28). 
Hence : 

*BA _ mrcQHA _ ^r_ __ ic 

1 ~ Q/T ~ 


By Galileo's theorem (p. 137), / EA may be replaced by J DA since 
EA and DA are chords of the same circle. This establishes the 
tautochronism of the cycloidal pendulum. 

Huygens did not include in the Horologium Oscillatorium his 
proof that in a cycloidal pendulum the restoring force is propor- 
tional to the arc of displacement. This short addendum is, 
however, important, since Huygens was the first to give the 
mathematical theory of simple harmonic motion. Leibnitz wrote 
to Huygens in March 1691 : " M. Newton has not treated the laws 
of the spring; I seem to remember having heard you say on an- 
other occasion that you have examined them, and that you had 
demonstrated the isochonism of the vibrations." We may sec 



for ourselves that the fact that the cycloidal motion is simple har- 
monic follows from Huygens's previous discoveries. He seems 
to have arrived at the conclusion about 1673. 

FIG. 29 

In Figure 09 the cycloid arc AC is divided at some point B. 
" Then," Huygens wrote, " the component of gravity along the 
plane tangential to the curve at A and the component at B is in 
each case proportional to the slope of the tangent ", i.e., in 
modern terms : 

Force down plane at A 
Force down plane at B 


g sin 






But, by a lemma to the theorem on page 139, Huygens showed by 
simple geometry that 

PC _ EC 

OC ~ PC' 

Hence : Force down plane at A __ EC 
Force down plane at B PC 

It was not difficult to show that EC is to PC as the lengths of the 
respective cycloidal arcs AC and BC. This relation may readily 
be demonstrated by the use of modern methods but Huygens's 


proof was not presented in a formal manner. With it the theorem 
on simple harmonic motion is completed : the acceleration at 
any point on a cycloidal arc is proportional to the length of the 
arc measured from the lowest point. It is thus clear from his note- 
book that Huygens cleared up most of the obvious problems 
which are raised by a study of oscillatory motion. Such prob- 
lems can only be dealt with satisfactorily by means of mathe- 

It must be admitted that Hooke, in England, earlier than 
Huygens, had commented on the conditions required for simple 
harmonic motion. Birch's History of the Royal Society (i 756-57) 
contains a passage by Hooke dated Nov. * i , 1666, on this subject : 
" The equality of duration of vibrations of differing arches or 
lengths depends upon the figure of the curve-line, in which the 
body is moved; which figure being for a very great part near 
the same with that of a circle, it follows that, the motion in 
differing arches of the same circle will be very near of equal 
duration/' Hooke's attempt to show mathematically that this 
followed from (Galileo's) mechanics was rather feeble, however. 
Even Brouncker, president of the Society and a much more able 
mathematician, made little headway with this subject. Yet both 
Brouncker and Hooke started where Huygens started and with 
the same fundamental conceptions. Hooke's Potentia Restitu- 
tiva, published in 1678, contained a further reference to the sub- 
ject and this work is usually regarded as being the starting point 
in the history of simple harmonic motion. It contains the law 
" ut tensio sic vis " and many deductions drawn therefrom. 
Nevertheless his contribution cannot be compared with that of 
Huygens, whose work was carried out quite independently. The 
properties of springs were well understood by Huygens in 1675, 
for this is the date of his invention of the spiral-spring regulator 
for watches. A further illustration of his grasp of the essential 
condition may be found in his experiments on the use of a tri- 
filar pendulum. This consisted of a heavy flat ring suspended 
by three equal vertical cords from three equidistant points. The 
oscillations of this pendulum were of a torsional character and 
he hoped that it would prove superior to the spiral spring, especi- 
ally in regard to the effects of temperature. It did not appear 
to matter how the restoring force was set up whether through 
gravity, elasticity or magnetic attraction he remarked. 


The Horologium Oscillatorium. Part Three 
Evolutes and the Measurement of Curves 

The occasion may be taken for a brief mention of Huygens's 
mathematical work. This seems to be divisible into two parts. 
In one part of his mathematical work Huygens may be said to 
have belonged to the classical schools of Archimedes, Hippo- 
crates of Chios and Eudoxus rather than to the period of Des- 
cartes, Newton and Leibnitz. He was, for example, greatly inter- 
ested in the three great problems of antiquity, those of squaring 
the circle, trisecting an angle and duplicating a cube, and he 
was in the classical sense the outstanding geometer of his age. 
The impress of classical antiquity lies on most of his early work. 
His earliest published work, Theoremata de quadrature hyper- 
boles ellipsis et circuli, ex data portionem gravitatis centra (1651), 
was undoubtedly inspired by Archimedes' work in hydrostatics 
dealing with the flotation of certain geometrical figures. It was 
in a sense a continuation of Archimedes' De /Equiponderantibus, 
and in finding the area of a given segment of a hyperbola, 
ellipse or circle, less than the whole figure, Huygens used the 
classical method of exhaustions. In the form of the " method 
of indivisibles " this method had been extended by Kepler and 
Cavalieri, the latter's work being especially influential. Huygens 
took the opportunity of showing up the fallacy in some work 
on quadratures by Gregory de St. Vincent, a contemporary Jesuit 
mathematician who appears not to have understood the newer 

Huygens began his treatise by determining limits to the area 
of a given segment of each of the above figures. He then located 
the centres of gravity and worked out some theorems, in particu- 
lar a theorem concerning a relation between the length of an 
arc of a circle, its chord, the radius and the distance from the 
centre to the centre of gravity of the segment. This paved the 
way for an examination or " Excursus " criticizing de St. Vin- 
cent's work. This excursus occasioned a dispute which lasted for 
ten years and in which several mathematicians were involved. 
It has not left any important results, however. 
* H5 


In January 1 652 Huygens began to study problems which lead 
to equations of the third or fourth degree. These " solid " prob- 
lems also arose, very probably, from a study of Archimedes' 
writings. In the course of his work Huygens (1654) carried out a 
new determination of the circumference of a circle. In the form 
of a determination of *, van Ceulen, early in the century, had 
made a new contribution to this ancient problem. Snell, in his 
Cyclametricus of 1621, narrowed the limits set by Archimedes in 
his original investigation but made use of propositions which 
were not rigorously proved. In all this, however, there was little 
if anything which could be called new in Huygens's work. The 
more interesting parts of his studies on the problems bequeathed 
by Archimedes are concerned with physics rather than with 
mathematics. In continuing the latter's studies of flotation, for 
example, he was able to make use of his fundamental principle 
that the centre of gravity of a system takes up the lowest posi- 
tion consistent with the restraints. In the first four theorems of 
his De Us quae liquido, supernatant (1650) he deduced in this 
way the horizontal surface of a stationary liquid, the equilibrium 
of floating bodies when the density is equal to that of the sup- 
porting fluid, and the celebrated law of Archimedes for the case 
in which the density of the solid is less than that of the liquid. 
In the other part of his work to which we may now turn 
Huygens was interested less in continuing the study of classical 
problems than in new and original developments. Reference has 
been made on more than one occasion to Huygens's fruitful in- 
terest in the cycloid. This curve may be taken as a starting point 
in reviewing briefly his important work on evolutes. Descartes 
influenced Huygens very much in this connection, for he had 
always insisted that any curve whose mode of generation could 
be clearly conceived belonged to Geometry. Descartes accord- 
ingly took all curves formed by the intersection of two moving 
lines, the rates of movement of which had a known ratio. 

Huygens did not commence his own researches on the 
cycloid until twelve years after Mersenne, in 1646, gave him his 
first information about publications on this interesting curve. 
Pascal's "Dettonville" problems, as has been mentioned, aroused 
Huygens for the first time. The problems were to find the area 
of a half-segment EBF of a cycloid (Fig. 30), the position of its 
centre of gravity and the volumes of the solids produced by 
revolution of the segment about BF and about EF. 


Huygens at first found the area EBF and thence the area of 
the whole segment EBO and he found the distance of the centre 
of gravity of the segment from the base EO and deduced the 
volume of the solid of revolution about this base. Pascal then 
suggested the calculation of the centre of gravity of the half- 
solid of revolution of ABD about AD. Huygens obtained a par- 

A D 

FIG. 30 

tial solution to this but found the work so difficult that he 
doubted if all Pascal's problems could in fact be solved. In pass- 
ing it may be noticed that Huygens improved on the ordinary 
proof required for the method of drawing a tangent to a cycloid. 
In the preceding figure a tangent to the cycloid at E must be 
parallel with the chord BG of the generating circle an interest- 
ing property which was given in Schooten's edition of Descartes's 

Pascal's proofs for the Dettonville problems compare favour- 
ably with the work carried out by Huygens in being altogether 
more elegant and at the same time more general. Wallis, how- 
ever, complained about Pascal's methods in his De Cycloide of 
1660, in which solutions to the Dettonville problems were 
obtained by means of Wallis's " arithmetica infinitorium ". A 
dispute between Wallis and Carcavy ensued, Huygens acting as 
intermediary, but it does not seem to have been of great signi- 
ficance. The really interesting feature of the whole episode is 
that it started Huygens on his study of evolutes. 

The idea of the evolute of a curve may be explained by refer- 
ence to the parabola. This curve (Fig. 31) may be described as 
the locus of points equidistant from a given point F (the focus) 
and a given line XY (the directrix). With the exception of the 
circle, all curves show varying curvature; whereas all normals to 
the circle intersect at the centre, the normals to other curves 
intersect at a series of points which generate another curved line. 



FIG. 31 

This is termed the evolute. This 
will be clear from the figure in 
the case of the parabola. 

The evolute may in fact be 
defined as the envelope of the 
normals to the given curve. 
Huygens no doubt saw, in his ex- 
periments with a simple pen- 
dulum, that the bob could be 
made to describe a variety of 
different arcs according to the 
shape of the curved " cheeks " 
between which it was suspended. 

Propositions 5 and 6 of the third part of the Horologium Oscilla- 
torium contain the discovery that the evolutes of a cycloid are 
themselves cycloid arcs. The proof is of course geometrical. In 
Fig. y. the arc AF is equal in length to the arc AC. By establishing 
this Huygens was able to rectify the curve and show that a cycloid 
is four times the length of the diameter of the generating circle. 
It is not considered necessary to recapitulate his method here. 

Huygens is remembered as the discoverer of the evolutes of 
a cycloid, but it must be pointed out that he dealt also with the 
evolutes of a parabola (proposition 8), of an ellipse and of a 
hyperbola (proposition 10). He also showed how to rectify 
curves for which the evolutes are known. This work has not been 
given much attention, possibly because of the more important 
general methods of quadrature worked out by Wallis. Huygens 


himself does not appear to have seen that the theory of evolutes 
would find its most useful application in a field in which he him- 
self was the first to explore: the theory of focal lines in geometri- 
cal optics. 

To the cubical parabola and cycloid Huygens later added the 
curve known as the cissoid to be included among curves whose 
rectification could be accomplished. The French physician, 
Claude Perrault, set the problem: to determine the path in a 
fixed plane of a heavy particle attached to one end of a taut string 
whose other end moves along a straight line in that plane. 
Huygens and Leibnitz studied this problem in 1693, and worked 
out the geometry of the tractrix. It was at this time that 
Huygens solved the problem of the catenary and determined the 
surfaces of certain solids of revolution. 

After his return to Holland in 1681 it seems as if Huygens 
resorted to pure mathematics and gave less place to physical work. 
About this time the infinitesimal calculus acted as a great stimulus 
to mathematicians, and it is not surprising, therefore, to find 
that Huygens's notebooks contain a great deal of work of a more 
analytical character done after his return to the Hague. A review 
of all his mathematical work would be a most valuable contribu- 
tion to the literature of the history of mathematics for, over a 
long period of years, from his earliest work on the circle there is 
a range of subjects, many of which bear a logical relationship, 
including work on maxima and minima, which links up with the 
work of Fermat and the English mathematicians of the period 
and culminates in the work of Leibnitz and Newton. 


The Horologium Oscillatorium. Part Four 
The Centre of Oscillation of a Compound Pendulum 

Passing reference was made on page 99 to the existence of action 
and reaction in the case of connected bodies. Galileo was unable 
to account for the behaviour of two small masses suspended along 
one thread (Fig. 33). He may have supposed that it should be 
possible for the masses to swing in unison, whereas in fact two 
independent vibrations exist in such a system. This is 
a difficult problem and there was no important practical 
reason for pursuing it. A more urgent problem was to 
calculate the period of a given compound pendulum, 
that is, a rigid body suspended so as to oscillate about an 
axis which passes through it. This is the type of pen- 
dulum employed in the pendulum clock, and the 
problem is to find /, the distance from the axis of oscilla- 
tion to the centre of oscillation. The latter is the point 
at which the entire mass would have to be concentrated 
in order to obtain the equivalent simple pendulum, that 
k is, that which possesses the same period of oscillation. 
_, The scientific principles of clock construction could not 

" ^ be said to be known until this central problem was solved. 
What was more important was that this particular problem opens 
up the whole subject of dynamical systems. 

Huygens was early acquainted with the problem by Mersenne 
and supposed that it emanated from him. The problem had, 
however, a longer history. The work In Mechanica Aristotelis 
Probletnata Exercitationis (1621), by Baldi, contained some 
erroneous suggestions on the centre of percussion and the sub- 
ject even seems to have been discussed by certain Greek writers. 
A modern illustration of the centre of percussion would be the 
point on a cricket bat at which the ball must be struck to secure 
the maximum effect with least effort. The centre of percussion, if 
it exists, is the same point as the centre of oscillation, a relation 
which was found empirically by Mersenne and proved theoreti- 
cally many years later. The interest shown by mathematicians like 
Descartes and Roberval in this problem must be attributed 


in some degree to the practical value of the subject in the 
design of sword blades. None of these, however, achieved any 
success. In a letter to Mersenne in 1646, Descartes dealt at some 
length with certain special cases of the centre of oscillation 
those of a long rod, a plane triangular figure and others. He 
clearly thought that the problem could be reduced to one in 
statics that of determining the centre of gravity of certain 
solids and planes. The work was surprisingly slipshod and 
amounted to little more than a series of ingenious guesses and 
no proofs could, of course, be given. 

Although Huygens's experiments on the centre of oscillation 
may have been carried out from as early as 1646, when his corre- 
spondence with Mersenne started him on the study of several 
problems of mechanics, his theoretical studies can hardly be 
dated earlier th^in 1659. He appears to have employed experi- 
ment much as Galileo did, and his procedure was in fact an 
excellent illustration of the inductive method. Starting with the 
simplest case of a linear rigid pendulum, he proceeded to study 
the oscillation of laminae oscillating in their own planes. For 
these cases he was soon successful in finding a way of calculating 
the length of the equivalent simple pendulum. Expressed in 
modern form this amounted to an application of the relation : 


where / is the moment of inertia about the axis of oscillation, 
m is the total mass and h is the distance of the centre of gravity 
from the axis of oscillation. The concept of the moment of 
inertia originated in Huygens's work, but it was given this term 
later by Euler. Huygens, however, discovered the important 
theorem which relates the moments of inertia of a lamina 
about two axes perpendicular to each other. All the laminae 
considered had regular shapes possessing an axis of symmetry. 
From these cases he passed on to the study of solids of revolution 
produced by rotating the laminae about the axes of symmetry. For 
this a method of evaluating Srar 3 for the relation / = 2mr 2 had 
to be devised and this part of the work makes the hardest reading 
for the modern reader who is accustomed to solve such problems 
by means of the integral calculus. 

As might be expected, Huygens started with certain problems 
left unanswered by Galileo and he was able to solve them by the 



application of his theorems on impact. As will be seen the 
subject of moments of inertia came out of the solution obtained 
in this preliminary work. 

D F 

FIG. 34 

In Fig. 34, an inflexible weightless bar carries two masses, D 
at the lower end, and E at some point in AD. Given the masses 
E and D, and the distances AD and AE, the problem was to 
find the centre of oscillation. The great difference here is that 
Huygens made the connection rigid, whereas in Galileo's 
problem the masses were connected by a light thread. 

Huygens's treatment was to suppose the pendulum with- 
drawn through an arc to the position ABC and then released. 
After passing through half of a swing the masses are again at 
the positions E and D and possess speeds which are different from 
those which they would acquire as the bobs of two separate 
pendulums of lengths AD and AE. Huygens supposed the 
two masses at this point to collide with masses which are respec- 
tively equal to the given masses but not connected. There is no loss 
of momentum and if the masses are all perfectly elastic the bodies 
F and G will acquire the speeds possessed by D and E, the latter 
being brought exactly to rest by the impact. Huygens then applied 
the equation connecting potential and kinetic energies to these 
masses. This manner of treating the problem was quite new. 
He supposed that the speeds acquired by F and G are such that 
they can rise against gravity to the positions N and V. The 
essential point in this process was that the centre of gravity of 
the masses could not ascend beyond its original height when the 
pendulum was in the position ABC. 


In Fig. 34 is shown the equivalent simple pendulum HK of 
length x. If we put AD = a t AE = b t the mass of D = m l9 
that of E = m a and their speeds at the lowest points respectively 
v l and z>2 we have 

speed of D v l _ arc CD AD _ a 
speed of K = V = 5rc?K * HK " x ' 
where v is the speed of the bob K through its lowest point. If 
the height CS is put equal to d, the height QP is given by 

QP = d.Z 

It has to be borne in mind that the equivalent pendulum swings 
with the same period as the compound pendulum. The height 
RN to which the masses F can ascend is such that 


Now # a 

Hence the height which the mass F can attain is ad/x and for 
the height MV, for the other mass, we obtain in the same 
manner b 2 d/ax. 

The centre of gravity of the two masses will then attain a 
height which can be calculated quite simply: the work done in 
raising the masses through their respective elevations NR and 
M V is equated to the loss of potential energy in the descent of 
the masses through CS and BO. These heights may then be ex- 
pressed in terms of x, the length of the equivalent simple 

F.NJR + G.MV = D.CS + E.BO 

ad b'd , bd 

or m l ~ + m a = m^d + m * 
x cix ** 

i.e m^d + m^b 2 d = m^dax + m^bdx. 

If this treatment is extended to a uniform rod, considered as 
composed of contiguous masses, the general formula 



is obtained. The quantity Smr* was thus introduced into 
mechanics. In this work Huygens made effective use of the 
idea of work done against gravity, as can be seen. Unfor- 
tunately, his ideas were not made explicit and he did not for- 
mulate a general method. His published work, however, .showed 
some improvement on this original form of the " direct " method. 
The device of supposing an impact with equal masses which are 
not connected was abandoned and, in the Horologium Oscil- 
latorium, Huygens simply supposed that the separate masses 
constituting the pendulum were freed from their connections at 
some point in the swing. The mathematics remains unchanged. 
It is important to appreciate exactly what Huygens had done 
in this piece of work, for it has a bearing on the development of 
the concept of energy in physics. The central idea lay in com- 
bining his fundamental principle that the centre of gravity of 
the masses cannot ascend as a result of a displacement occurring 
under gravity with Galileo's relation connecting the speed 
acquired in falling with the square root of the height. In the 
Horologium Oscillatorium the speeds of suspended particles in 
any point in the path were compared with the corresponding 
heights of descent, a procedure which really amounted, in com- 
bination with the conservation of vis viva, to the application of 
the law of conservation of energy in mechanics. It is possible 
from this starting point to obtain an equation showing the con- 
stancy of the sum of the kinetic and potential energies for an 
isolated system, namely : 

T + V = H 

in the form given by Lagrange. If any doubt existed as to the 
significance of Huygens's contribution to the subject of energy 
it may fairly be stated to be removed by a later statement of his 
ideas. In his MS. of 1693, two years before his death, he wrote: 
"In all movements of bodies whatsoever, no force is lost or 
disappears without producing a subsequent effect for the produc- 
tion of which the same amount of force is needed as that which 
has been lost. By force I mean the power of raising a weight. 
Thus, a double force is that which is capable of raising the same 
weight twice as high." The word for force in this passage was vis, 
a word which, like potentia, was used in the seventeenth century 
in the two senses of force and energy. Unfortunately, the defini- 


tion is ambiguous but it would fit the notion of work or energy 
rather better than that of force in the Newtonian sense. The 
passage clearly reduces to this: there is something about a 
moving body which enables it to effect changes in the state of 
either itself or other bodies, and these results are quantitatively 
related with the cause which is the force or energy involved. The 
same idea was far less clearly expressed by Leibnitz /about this 
time : "... it seems necessary to admit in bodies something 
other than magnitude and velocity unless we are willing to deny 
to bodies all power of action ". 

To return from this digression to the contents of the 
Horologium Oscillatorium : Huygens found it impossible to 
apply the " direct " method to determine the centre of oscillation 
of suspended solids. The location of the centre of oscillation of 
a suspended sphere, for example, cost him much time and the 
method given in this work is long and difficult. His ideas appear 
altogether strange to the modern reader and an explanatory note 
will not be out of place at this point. To understand how 
Huygens's ideas took shape it is of interest to know that he read 
a work entitled Tractatus Physicus de Motu locali by one 
Mousnier, a pupil of Pere Honori Fabri, which came out in 1646, 
and in this there was a novel attempt to solve the problem 
of the centre of percussion in the general case. Little seems to 
be known of Mousnier, and his ideas may, to a large extent, have 
been derived from Fabri. Mousnier used a concept which he 
termed the " impetus " of motion. By this he meant a quantity 
varying as the product my for a particle in motion and he applied 
this to the elementary parts of an oscillating body as follows : 
considering the oscillation of a plane surface about an axis in its 
own plane, to evaluate the total " impetus " of all the elements, 
Mousnier took lines perpendicular to the surface and these 
represented by their lengths the speeds of the elements (to which 
they were normal) at the lowest points in their paths. This con- 
struction generated a wedge-shaped solid. For laminae of simple 
form the volume of the wedge could be calculated. Mousnier 
did not know, as Huygens did, that the centres of percussion and 
oscillation are identical. The identity had not been proved at 
that time and it was doubted by some, notably by Roberval, who 
thought the two points were only approximately in the same 
position. Proceeding by experiment, and at first inductively, 
Huygens was able to find the equivalent simple pendulum 


for a lamina oscillating in the way stated, and this opened the 
way to deal with suspended solids. The problem in every case is 
to evaluate the expression Smr 2 . 

To take a simple case, the lamina ABC (Fig. 35) oscillates 
about the axis EAE which is tangential. Huygens proceeds 
to construct a wedge-shaped solid 
on the lamina as base : a second 
plane is taken at 45 and a 
generating line DB, perpendi- 
cular to ABC, moves round 
the boundary of the lamina 
tracing the projection on the 
inclined plane. The centre 
of gravity of the enclosed solid 
can be found if the base is of 
simple geometrical form, and a p IG - 

plane of symmetry meets the 

axis of oscillation perpendicularly at A. From the centre of 
gravity, X, Huygens drops a perpendicular XL to the base and 
gives the name subcentric to the straight line AL. This is only 
a term for the distance of the centre of percussion from the axis. 
The wedge is in fact a geometrical representation of the ex- 
pression fv.da where da, in the ordinary calculus notation, is an 
element of the area of the lamina and v is its linear velocity at 
the mid-point of the oscillation. If we write dm in place of da it 
becomes more clear that the wedge represents the summation of 
the linear momentum of the lamina as it passes through the 
mid-point of its swing. By determining the point L, Huygens 
was in fact finding the position at which the total mass M would 
possess the same momentum. Putting AL = /, we may, for 
example, take the simple case where, for the lamina, we have 
simply a uniform rod. The result for this case is already known 
by the direct method to be 

/ = 

jydy fcy 

(this is a modern form of the expression given before). The same 
result may also be obtained by the new geometrical method, for 
the point X would obviously lie at the point of intersection of 
the medians of a right-angled isocceles triangle. It follows from 
this that AL = 4B(Fig.36). 


In the case of a lamina a method of finding the volume of 
the wedge was required. Huygens showed that this could be done 
by means of a simple relation which may be expressed : 

volume = (area of baseXdistance AF) 
where F is the centre of gravity of the lamina (Fig. 37). Huygens's 

FIG. 36 

FIG. 37 



method is in effect to treat the determination of Srar 2 about the 
axis EAE as a geometrical problem. He does not explain how he 
made his discoveries but shows how 2mr 2 for a lamina of simple 
shape can be found for several different axes of oscillation 
including an axis through the centre of gravity or at a fixed 
distance from it. 

When he took up the problem of 
finding the moment of inertia of a sus- 
pended solid his procedure showed an 
essential resemblance with that employed 
when such problems are solved by means 
of the calculus. In this case the regular 
solid is divided by planes chosen in a 
suitable direction so as to form a series 
of laminae of simple geometrical form. 
The expression y*.da is then integrated 
between the limits of y, the distance of 
any element from the axis of oscillation 
after expressing da in terms of the co- 
ordinates x and y . The integration is then 
required for an expression in terms of y 
and dy. This is scarcely a satisfactory 
statement about Huygens's method, FIG. 38 



especially since he lacked a general method of summation, but it 
does not appear possible to describe his procedure in greater detail 
and the reader who is interested is referred to the German transla- 
tion in Ostwald's Klassiker der Exakten Wissenschaften, No. 192, 
to which are appended extensive notes in more modern form. 
Huygens's method could not very well be applied to more than 
a few regular solids. 

The centre of oscillation C is in all cases lower than the 
centre of gravity G of the suspended solid. If / is the distance of 
the former from the axis of oscillation O and r the distance of 
the latter, Huygens showed how (/ r ) could be calculated in 
certain cases. He was impressed by the discovery (1664) that the 
product (/ r )r is constant and he later termed this the 
" rectangulum distantiarum". For parallel axes of oscillation 
the equation : 

(/ - r )r = (V -r > ' (i). 

could be applied, the second set of symbols referring to the 
second position of the pendulum. This enabled him to calculate 
exactly the effect of lengthening the suspension. If on the 
second occasion the axis of oscillation is made to pass through 
the point which was formerly the centre of oscillation it follows 
that the new centre of oscillation will be at the position of the 
former axis, for, from Fig. 38 : 

'.' = I - r Q (ii). 

and since I r _ r/ 

/'~-~~V ~ r 

I' r.' = r 

and from (ii) /' = r + I r 

= /. 

This means that the compound pendulum has the same period 
in these two positions. The idea of the reversible pendulum was 
of course applied very successfully by Kater. The Kater 
pendulum is in fact the best laboratory method for the deter- 
mination of the acceleration of gravity, since the distance 
between the two knife edges on which its period is the same can 
be accurately measured. 

No theorem requisite for the complete theory of the 
pendulum clock was omitted by Huygens. The initial difficul- 
ties being cleared away he completed his treatise with a consider- 


ation of the effect of moving a small weight or rider along the 
rod of the clock pendulum. The latter consists of a rigid rod 
carrying a heavy sphere at its lower end. To find the centre of 
oscillation of such a pendulum it was necessary to combine the 

values of - -- for the rod (of length L) and the sphere found 

separately. Taking the respective masses to be m x and m a he 
showed that Smr a = ^rnJJ for the rod and 2mr 1 = m J L a for the 
sphere, L being measured from the centre of the sphere. The 
corresponding values for 2mr were ^m^L and mJL. Hence for the 

A simplification is introduced into this work in that the radius of 
gyration of the suspended sphere is not L but (i/jR* + L*^ 
where R is the radius of the sphere. Huygens proceeded to 
calculate the centre of oscillation for the same pendulum carry- 
ing a small spherical rider at a given position on the rod. He 
showed that for a given alteration in the period there were two 
positions in general for the rider. 

At the end of the work, Huygens proposed a unit of length 
based on the pendulum. The standard foot, or pes horarium, 
was to be one-third of the length of a simple pendulum which 
beat seconds at Paris. The size of the bob would, of course, be 
immaterial since for a sufficiently large bob, which could not be 
regarded as a simple pendulum, the centre of oscillation could 
now be found. The objections to this unit were the variation of 
g, the acceleration of gravity, with latitude and the inaccuracies 
which must arise in measuring the length of the thread. It was, 
however, an ingenious suggestion designed to overcome the 
objections which apply to the dependence on a bar of standard 

In spite of its forbidding appearance in the original, 
Huygens's work in this part of the Horologhim Oscillatorium 
contains some strikingly original ideas. The idea of work or 
energy, implicit in some of his work on impact, turns up again 
here in a more precise form. The idea of vis viva indeed comes 
from the work on impact while that of work done is involved in 
the direct method of finding the centre of oscillation. Someone 


indeed was bound to derive the last idea from Galileo's equations 
of motion which, multiplied by m give (in modern symbols) : 

mv = mgt = Pt (i). 

ms = imgf = IP? (ii). 

mgs = \mv* (iii). 

If Huygens's work had been more promptly published, and 
still more, if he had not cast all he wrote in geometrical form, the 
third equation might sooner have entered into mechanics. In 
the work on moments of inertia, as in the contribution to the 
subject of energy, Huygens has not been duly recognized. Like 
Newton's Principia, the Horologium later came to be regarded 
as pretty hard going for readers who were accustomed to analy- 
tical methods. 


The Cause of Gravity 

The idea of a force which acts towards the centre of the 
Earth dates from the earliest times 1 . Aristotle's doctrine that 
the elements, excepting fire, have a tendency to take up their 
" natural " places was a modified form of the idea and it was 
reproduced by the scholastic commentators. A clearer statement 
was given by Copernicus, who wrote " The Earth is spherical, for 
all its parts strain towards its centre of gravity." Much later 
Mersenne defined the " centre of the universe " as the point 
towards which all heavy bodies tend in straight lines. Gilbert 
(1600) attributed the action of gravity to other bodies besides the 
Earth, but he did not regard it as universal. The reason for 
motion under gravity was for him " a substantial form, special 
and particular, belonging to the primary bodies," which suffici- 
ently indicates the influence of Aristotle. Kepler first considered 
gravity to be " a mutual attraction between parent bodies which 
tend to unite and join together ". It is, he said, the attraction 
due to the earth rather than a tendency in the stone which 
causes the latter to fall. 

Up to Huygens's formulation of centrifugal force it was a 
real problem for the early seventeenth century scientists why 
the earth and moon were not attracted into contact with each 
other. Borelli, in Theorias Mediceorum plonetarum ex causis 
physicis deductse (1665) supposed that there was a tendency for 
the two bodies to come together, but that this was prevented by 
some kind of fluid pressure. Following Kepler, Borelli believed 
that the sun emanated some sort of " virtue " which kept the 
planets moving in their orbits. It is a most curious fact 
that although his theorems on centrifugal force were discovered 
about 1659, very probably Huygens did not appreciate the im- 
portance of the work in relation to the idea of a physical basis 
for the solar system. The possibilities of uniting the idea of 
gravitational attraction with that of centrifugal force were 
realized in England by Halley and, of course, by Newton. But 

i The history of the subject has been very well outlined by Duhem: La 
Thtoric Physique, p. 370-414 (1906). 
L 161 


Huygens was debarred from his due place in this most important 
development by certain unfortunate preconceptions. There is 
otherwise no obvious reason why he should not have forestalled 
Newton by many years in regard to this part of the Principia. 
The influence of Descartes here led Huygens to adhere to the 
hypothesis that all change is brought about through physical 
contact between bodies, either directly or through the medium 
of some subtle matter which filled the intervening space. 
Following Descartes he took an approach which bade to reduce 
phenomena to kinetics. Besides this, however, he realized the 
importance of structure or the " conformation " of physical 
bodies as an explanatory principle. But change effected across 
empty space seemed to him to be remote from experience and to 
leave a gap in the cause-effect sequence. Using the idea of force 
as the scientific mode of cause, Huygens considered it necessary 
to restrict the term to the operations of bodies on each other 
perhaps under the influence of Descartes 's dictum that what is 
true may be clearly conceived. It was this limited view of what 
can constitute a causal mechanism that led Huygens to postulate 
his various " media ". For the transmission of light he required 
an ether, for magnetic fields, a magnetic medium, and for gravi- 
tational effects a " subtle matter ". In the Traite de la Lumicre 
the situation is by no means simplified. There is a possibility, he 
there remarks, " that the particles of the ether, notwithstanding 
their smallness, are in turn composed of other parts, and that 
their springiness consists in the very rapid movement of a subtle 
matter which penetrates them from every side, and constrains 
their structure to assume such a disposition as to give to this fluid 
matter the most overt and easy passage possible ". Gassendi's 
atomic doctrines clearly influenced Huygens. As against Boyle, 
Huygens did not consider that the motion of the atoms of 
ordinary matter was sufficient to explain elasticity and thermal 
expansion. Such effects, he wrote, " cannot be explained without 
supposing the same subtle matter in motion with an extreme 

The most interesting remark on the subject of gravitational 
force to be found in Huygens's writings has been quoted on page 
1 1 8. In addition there are two notes in his MS. of 1668 and 1669, 
which are as follows : " Gravitatem sequi quantitatem materiae 
cohaerentes in quolibet corpore " and " l poids de chaque corps 
suit pr6cisement la quantit de la mature qui entre dans sa com- 



Drawing by Huygens of his Vacuum Pump of 1 668 


position ". Huygens, then, made the distinction between mass 
and weight before Newton, but these statements do not imply 
that Huygens regarded gravity as something inherent in matter. 

After 1661 Huygens was occupied with experiments with a 
simple air pump. His interest in this instrument was aroused 
during his visit to London. In 1668, an improved pump was 
constructed. This machine had a brass piston bound with fine 
flax in place of a wooden piston impregnated with wax. The 
pump (see Plate VI) was later described by Denis Papin in 
Nouvelles Experiences du Vide (1674). Actually Huygens and 
Papin in their experiments of 1674-7 added little to the work of 
Boyle; Huygens was not sufficient of a chemist to pursue the 
more interesting problems. The observation which most 
attracted interest was his discovery of the non-descent of 
columns of water from inverted tubes placed in the evacuated 
space of the vacuum pump receiver. Much correspondence 
arose over this anomalous observation which Huygens himself 
explained by assuming the existence of a subtle matter which 
exerted a pressure even after the pressure due to the air had been 
almost entirely removed. The effect was only noticed with water 
(and later with mercury) which had been " purged of air ". As 
has been noted in the earlier part of this book, Wallis doubted if 
Huygens's subtle matter really explained the phenomenon: "For, 
if this Matter be so subtile as to pass, through the top of the 
Glass, upon the Quicksilver ... I do not see, why it should not 
balance itself (above and below) in the same manner as Common 
Air would do, if the Tube were pervious to it at both ends, and 
the Quicksilver, by the preponderance of its own weight, fall pre- 
sently." Hooke, although a Cartesian, came nearer to the truth. 
In his Micrographia he mentioned the cohesion between liquids 
and glass. Mercury and glass were too different in their natures 
to cohere but water, " being somewhat similar to both, is, as it 
were, a medium to unite both the glass and the mercury to- 
gether ". In the absence of water, however, Hooke felt obliged 
to accept Huygens's explanation. Even Newton allowed that 
Huygens's explanation was the probable one. t Newton also sug- 
gested that the rise of liquids in capillary tubes might be due 
to the ethereal medium but his own adherence to the hypothesis 
of an ether was, as is well known, somewhat inconsistent. 

His persuasion that a subtle matter exists was a strong argu- 
ment for the retention of Descartes's vortices by Huygens. In 


1667 he attempted to work out a satisfactory explanation of gra- 
vity as an effect of circular motion. His hopes that this would 
prove possible no doubt rose as the existence of a subtle matter 
became more and more accepted by men of science. By 1669 he 
felt himself to be in a position to put his view before the 
Academic Royale, the occasion being a discussion on the subject 
of gravity in which the other speakers were Roberval and Mari- 
otte. In fact, no other theory was put forward and the discussion 
became a criticism of Huygens's theory. 

Huygens proposed to limit himself to terrestrial gravity and 
for this purpose considered the Earth to be an isolated system. 
The Cartesian vortex moved, according to his view, around the 
Earth in such a way that the subtle matter everywhere moved 
parallel to great circles on the Earth's surface. So far as one can 
judge there was no question of gravity extending to the moon. 
To illustrate his argument Huygens described an experiment 
in which a bowl of water is rotated about its axis. Heavy particles 
introduced into the rotating liquid were found to be propelled 
towards the centre as the rotation slowed down. Huygens pro- 
posed that the subtle matter which played such a part in his 
vacuum pump experiments was in fact the matter of the vortex 
about the earth. If the speed of rotation were high enough this 
would account for gravity as a centripetal reaction. But it was 
necessary to suppose that the circular motion of the vortex was 
" natural " and not constrained. The subtle matter, Huygens cal- 
culated, would have to be in rotation with a speed about seven- 
teen times as great as that of the diurnal rotation of the earth. 

To all this Roberval and Mariotte made the more obvious 
objections, for it is difficult to conceive how an ether which is per- 
meable to matter can exert a pressure on matter. Going further, 
they questioned the validity of restricting all explanations to the 
terms of matter and motion. They questioned the evidence for 
supposing that circular motion was in this instance " natural ". 
Roberval preferred the view that gravity is a mutual attraction 
between the particles of bodies. He had, he said, maintained this 
view as early as 1636. Both he and Mariotte considered that Huy- 
gens had only replaced one mystery by another. 

Huygensfs Optical Studies 

The study of the propagation of light and its behaviour at 
reflecting and refracting surfaces has a long history. This is, in- 
deed, the oldest branch of physics and it is necessary to recall 
that for centuries the subject possessed for many minds a certain 
mystery which was dispelled only as the phenomena came to 
be seen as illustrations of general laws. Descartes put the sub- 
ject of light in a central position in his natural philosophy; one 
of his works was entitled Le Monde ou Traite de la Lumiere. 

Yet Descartes achieved comparatively little in his study of 
light and it was Kepler's Dioptrice of 161 1 which, more than any 
other single work, laid the foundations of modern optics. 
Euclid's Optics (c. 300 B.C.) contained a statement of the equality 
of the angles of incidence and reflection for a plane surface and 
Claudius Ptolemy (c. A.D. 150) introduced the study of refraction. 
Kepler spent much time on but failed to discover the relation 
between the angles of incidence and refraction. His most impor- 
tart relation was in fact (in modern symbols) 


i - r 


Here i is the angle of incidence and r the angle of refraction and 
D the angle of deviation (i - r). That is, the deviation varies as 
the angle of incidence a relation which is nearly true for angles 
of incidence less than 30. Kepler showed that the constant for 
ordinary glass was about i. He then calculated the principal foci 
in the cases illustrated : 

(i) Parallel rays incident on a convex glass surface : 




He obtained the result f = yr where f is the distance of the prin- 
cipal focus F and r is the radius of curvature : 

IAJ- i ' 0.5 

(ii) Parallel rays incident on the inside surface of a glass block 
having a convex surface: 

FIG. 40 

He obtained f = ar a result which follows simply from the mod- 
ern relation : 

/ - r ~ r 
(iii) Parallel rays incident on a convex lens: 

FIG. 41 

He obtained the result f = r for the case of a lens having two sur- 
faces of equal radii of curvature (r). This is correct, for : 

= r . 

Putting^ - -r 2 , f = 

5 0.5 */ 

There were, however, no general equations for the treatment 
of lenses up to the time when Huygens began his work. Cava- 
lieri, following the lines of Kepler's work, in 1647 proved the rela- 
tion used above for the focal length of a thin lens and Isaac 


Barrow in 1674 found by a geometrical method the image formed 
by a thick lens upon which an axial pencil falls. " Such cum- 
brous geometrical investigations involving the separate consider- 
ation of numerous particular cases/' writes Professor Wolf (A 
History of Science, Technology and Philosophy, I, p. -248), " were 
eventually superseded by the analytical methods of Descartes, 
which Halley, in 1693, successfully applied to the problem of 
finding the general formula of the thick lens." 

Descartes did more than Kepler to treat the problem of 
spherical aberration but in this and other respects he arrived at 
no useful result of practical importance. His recommendations 
concerning elliptical and hyperbolic surfaces were chimeras and 
much needl-ess labour was lost in attempting to put his ideas into 

Unfortunately Huygens's work in optics belongs to the earlier 
period referred to by Professor Wolf and his writings are very 
tedious to read because of the absence of algebraical formulae. 
Throughout his life he was constantly amplifying and re-writing 
his manuscript and only the Traitf de la Lumtire, considered in 
a later chapter, appeared during his lifetime. Of the rest of 
his extensive researches some were made public in lectures to the 
Academic Royale but the remainder was hidden until published 
in 1703 by which time it had really ceased to possess more than 
historical importance. 

Refractive indices are not widely quoted by Huygens in his 
Dioptrica. He seems to have supposed, from the small number 
of materials then available, that exact values were unimportant. 
His method, in the case of glass, was to determine the focal length 
of a plano-convex lens and to apply the formula 

The method of finding ^ from the true depth and apparent 
depth of an object seen through a rectangular block was also 

FIG. 42 


known to him and was used in his researches on Iceland spar. 
For liquids his method was to fill a large glass cylinder and find 
tfye distance of the focal line when the incident light, perpendicu- 
lar to the axis, was parallel. 

For a cylindrical lens we have : 

Putting f=v'-r (see figure above): 


or 2JJO/ - pif = i(v' - r), 

u w/ - f 

whence p = -- - 

t; -- 

as given by Huygens. Huygens, however, arrived at this result 
by purely geometrical methods. 

Huygens's method of treating refraction may be illustrated 
by the problem of finding the principal focus of a convex 

FIG. 43 

spherical surface. Huygens showed that if C is the centre of 
curvature and NP, OB are parallel with the axis AQ, 

taking ~^ = (x then Q is the point through which the rays will 

pass. This, of course, is correct, for, using the formula : 

i* _ i __ IA- i 

v ~u ~~ r 

and putting u = oo, v = /, 

a UL-I , -/ 

_!_ = L. whence JJL= L- 

f ' ' *-f 


Needless to say, no formula of the type quoted is used in the 
work. But the correspondence of Huygens's geometry with 
modern practice can be perceived if his diagrams are carefully 
examined. His method of locating the image of a point source 
produced by a lens is to employ the relation 



DC ~~ DP 
DO.DP = DC 3 

Now, DO = u-f, 
i.e., uf + uf=uv, 

and hence (t< - /)(w + 1;) = 


which at once appears familiar. The original proof is, however, 
too long to quote. 

Huygens pointed out that there is an optical centre in a lens 
such that rays passing through emerge after traversing the lens 
parallel with their original directions. In Fig. 45, E and F are the 

f P 

FIG. 45 

centres of curvature of the lens faces and ED, FB are radii. The 
point L can be found since it may be shown that 


LD = ED 


Some consideration was given to the effect of a lens immersed 
in a liquid such as water. Huygens showed how to calculate the 
refractive index for the two media in contact, knowing the in- 
dices with respect to air. He also gave a useful account of the 
human eye. He distinguished the liquids known as the vitreous 
and aqueous humours but made the mistake of supposing that 
the content of the crystalline lens was also a liquid. In consider- 
ing the location of an image at the least distance of distinct 
vision, a fundamental idea in the theory of optical instruments, 
his work was rendered unnecessarily complicated by an unfor- 
tunate choice of distances. Huygens measured the distance of 
the image from the eye and not from the lens through which 
it was viewed. However, he was led to discover an interesting 
theorem concerned with the magnification produced by a system 
of lenses. This, quite briefly, stated that by interchanging the 
positions of the eye and the object, without altering the posi- 
tions of the lenses, the object appears to the eye to be of the same 
size as before. This conclusion is of purely theoretical interest 
but there is a sequel : Lagrange later obtained equations which 
accord with this peculiar theorem and this in its turn led to work 
by Hamilton, Clausius and Kirchhoff. There seems to be a con- 
nection between Huygens's work and later developments of the 
conception of optical distance. 

Huygens's more important researches on lenses dealt with 
the subject of spherical aberration. It was known from the time 
of Kepler that the middle of a lens having spherical surfaces had 
not exactly the same focal length as the peripheral pan of the 
lens. To secure better definition it was customary to employ a 
stop covering all but the middle of the lens. The aperture used 
was judged by experience. Huygens saw that it should be pos- 
sible to calculate the aperture permissible for any given lens. 
It should also be possible to decide the optimum form for a 
lens of given focal length. So early as 1653 Huygens compared 
the distortions produced by a plano-convex lens first with the 
convex and then with the plane side towards the light. He also 
introduced the idea of optical thickness as measured not by the 
actual thickness at the middle but by the difference in thickness 
at the middle and at the edge. For a plano-convex lens the focal 
length of the peripheral part can be calculated in terms of the 
radius of curvature and the distance of the incident ray from 
the axis. For rays close to the axis the focal length is given by 


the ordinary formula. Huygens performed the calculation for 
the two positions of the lens and showed that the separation 
of the foci was less when the light was incident on the curved 
face. When the curved face is turned towards the light the rays 
suffer approximately equal deviations at the two refractions. 
Huygens saw that spherical aberration increases with the amount 
of deviation occurring. It thus appeared advantageous to use 
two lenses at the eye-piece of a telescope instead of only one 
and it could be secured that the total deviation was divided 
equally between these two lenses. Huygens showed that this 
was so when the separation of the lenses was equal to the differ- 
ence of their focal lengths. 

Huygens considered that the Dutch, or as it is sometimes 
called, the Galilean, telescope could be made more free from 
spherical aberration than the Keplerian telescope which em- 
ployed a convex eye-piece. Seeing that the concave eye-piece of 
the former compensated in some degree for the aberration of the 
objective, a much greater aperture should be permissible and 
thus a greater magnification might be obtained without increase 
of length of the telescope. Huygens recognized, however, that the 
Dutch telescope suffered from having too narrow a field of view 
for astronomical purposes. Various means of remedying this were 
considered but they were abandoned because of the difficulty 
of grinding lenses to a specified form. 

It was still essential, however, to define the practicable limits 
to the aperture for any given lens, for there continued to be a great 
deal of confusion on this point. Huygens considered definition 
to depend simply on the quantity of light per unit area falling 
on the retina of the eye. His procedure was to start with a tele- 
scope of known dimensions which gave good results and to cal- 
culate the lengths and apertures for others of the same standard 
of definition. For this purpose he limited himself to a consider- 
ation of the objective and ignored the eye-piece. 

If f and f f are the focal lengths of two objectives and <f> and ^' 
those of the corresponding eye-pieces and g and g r the linear mag- 
nifications, d and df the diameters of the apertures, then, for 
equal intensity of light at the eye : 

d _ j _ f *L 

d' ~ g> ~ ' f 

** - *'* (\) 

f ~ T ( 



To compare the aberrations we may suppose, with Huygens, 
that the lenses are of the " same sort ", i.e. that 

where R represents the radius of curvature of a lens surface. In 
Fig. 46 the distance FFj represents the separation of the foci. We 
may put FF 1 = &, F'F/ = $'; then it may be shown that : 

a ff ,.., 

__ _ '__ (n) 

9' ~ d'*f .............................. () ' 

FIG. 46 

From the figure it is clear that the rays through the peripheral 
part of the lens will meet the focal plane through F at points H 
on the circumference of a circle of radius FH. FH is given by 

FH=& tan 6. 

The focal plane of the objective is also that of the eye-piece of an 
astronomical telescope. The image of the circle will therefore be 
the circle of aberration for the telescope and its radius will be 
given by 

8 tan 
X where K is a constant. 

Putting tan 6 = -7. this becomes K - . For both telescopes 

2/ 2/0 

to produce equal circles of aberration 



or by (ii) 
and by (i) 






\ J 


These equations, derived after the manner given in the intro- 
ductory notes to volume 13 of the Oeuvres Completes, summarize 
the rules elucidated by Huygens. 

As has been mentioned, the existence of spherical aberration 
was not unknown to Huygens's contemporaries nor to some of 
his predecessors. Maurolycus (in 1553) even mentioned it. New- 
ton, in treating the aberration of a plano-convex lens with the 
plane side towards the light found the value of the aberration, 

FF l in the form of a series in which the first term was 

- i 

where e was the thickness of the lens and JJL the refractive index. 
Picard also deserves mention for, in his Fragments de Dioptrique, 


he obtained a value of ~ for the spherical aberration of a glass 

plano-convex lens receiving light on the convex side. His method 
resembled that of Huygens. Molyneux gave a single numerical 
calculation to his Dioptrica Nova (1697); he concluded that the 
" depth of focus " is smallest in the case of a plano-convex lens 

FIG. 47 

when the rays are received on the convex surface. Huygens spent 
much time on the relation between spherical aberration and the 
inclination of the lens surfaces at the periphery. He saw that for 
a thin lens the deviation produced at a given point may be 
regarded as constant. Later he recognized that chromatic aberra- 
tion was the quantity which was most affected by variation in 
the angle of inclination. 

When he turned to the study of chromatic aberration, Huy- 
gens made use of the work of Newton. There appears to be no 
evidence that Huygens measured the refractive index of glass for 
different colours. Now Newton estimated the circle of chromatic 
aberration (radius CO in Fig. 47) to be one-fiftieth of the diameter 
of the lens employed. The fact that Huygens obtained divergent 
results may have been due to the use of a very different glass or 


to the choice of a different circle. Newton's circle was situated 
midway between F v and F r ; the choice of the plane through F v 
would give a circle of greater diameter. 

To compare the relative magnitudes of chromatic and 
spherical aberrations Huygens took a plano-convex lens of 
twelve inches focal length and allowed an aperture of half an 
inch. The optical thickness was thus only 1/192 inch. With the 
curved surface towards the light the spherical aberration 
measured along the axis is, by Picard's rule, 1/164 inch. The 
chromatic aberration may be calculated from the relation 
f r - f v cof where o> is the dispersive power of the glass and f is, 
of course, the ordinary focal length (for the mean ray). Taking 
o) to have the value .017 this means that the distance f r -fv is 
i /5 inch. This is 33 times as great as the spherical aberration. In 
large telescopes the difference between the effects would be even 
greater. Following the method previously used, Huygens esti- 
mated the aperture of any given lens by comparison with an 
instrument which gave satisfactory results. He showed that the 
ratio of the apertures of the lenses used for the objective and 
eye-piece would be equal to that of the square roots of the focal 

When Huygens used a telescope expressly designed for observ- 
ation of Saturn to study the moon he found the brightness of the 
image too great. He accordingly restricted the aperture of the 
objective very much but was surprised when a point was reached 
when the clearness of the image suddenly diminished. He con- 
sidered this was due to some property of the eye. " For also," he 
wrote, " when one places in front of the naked eye a plate having 
a hole of i/5th or i/6th of a line in diameter, the edges of 
objects begin to appear less clear, and the confusion becomes 
greater the more one diminishes the size of the aperture.' 1 Huy- 
gens does not seem to have identified the effect with that of the 
diffraction described by Grimaldi in his Physico-mathesis de 
Lumine (1665). It is also mentioned in Newton's Principia. 

Of course, certain of the conditions affecting the construction 
of telescopes apply also to the construction of micioscopes. Huy- 
gens was for a time a keen microscopist and he sided with Leeu- 
wenhoek, Redi and Swammerdam in the view that the evidence 
was all against spontaneous generation even of protozoa. Huy- 
gens treated the optical system of the compound microscope in 
much the same manner as he treated that of the telescope. The 


diagrams he gave to explain the principal features were of more 
ojf less modern form; they show the refraction of the rays so that 
a real image is formed by the objective. The image, as magnified 
by the eye-lens, is viewed at the least distance of distinct vision. 
He derived an expression for the magnification of the instrument 
and showed that if this is increased by decreasing the focal length 
of the objective there is an inevitable decrease in the depth of 
focus. As a practical microscopist he found that for certain observ- 
ations better results were obtained when the object reflected light 
than when the light was transmitted. He was responsible for 
the invention of dark ground illumination. 

At this point there is a temptation to introduce an account of 
Huygen's excursions into biology, if only to correct the impres- 
sion that he was so exclusively a student of the physical sciences. 
His studies of infusoria were quite notable and in one or two 
points, original, and of course the fact is that these seventeenth 
century men of science took no trouble to observe any artificial 
boundaries to their " subjects ". Huygens not only translated 
Leeuwenhoek's writings on microscopic observations, he was a 
fellow-observer who repeated and extended his compatriot *s 
experiments. Like Leeuwenhoek, he was opposed to the theory 
of spontaneous generation and his own experiments on the sub- 
ject are a prelude to those later carried out with such perfection 
by Pasteur. There is something immensely stimulating about this 
all-round activity in science, this feeling of the arresting interest 
of so piany and such diverse problems. But, in reading Huygens 's 
correspondence and in working at his notebooks, one has to clear 
away the notion of the " specialist " scientist. One sees that his 
interest in the microscope is as practical as his interest in the 
telescope was at all times; he wanted a science of optics in order 
to make his instruments, to perform calculations in astronomy, 
to understand the phenomena of " false suns " and haloes, to 
pursue the questioning of Nature out into the remotest spaces of 
the universe or down to the limits of the smallest living 

The Wave Theory of Light 

The notion that light is in nature akin to sound is very ancient. 
Thinking along these lines, Roger Bacon stated that light tra- 
velled in successive stages through the air; hft language hinted 
at some kind of vibratory motion. After that no progress beyond 
a general assumption of some analogy between light and sound 
was made up to the time of Francis Bacon, and he had little 
to suggest on this subject. Descartes first gave an interpretation 
of the facts, so far as they were known, which had some appeal 
for men of science in the seventeenth century. For him light was 
to be regarded as a pressure transmitted with infinite speed 
through the subtle matter which filled his universe. "Light in 
luminous bodies," he wrote, " is only a certain movement, or a 
very lively motion which passes towards our eyes ... in the 
same way that the movement or resistance of bodies which a 
blind man meets passes to his hand by the medium of his stick." 
Again : "... it is not so much the movement as the inclination 
to move of luminous bodies that we must consider as their light 
. . . the rays of this light are nothing more than the lines along 
which this inclination tends." 

Huygens was not influenced exclusively by Descartes, how- 
ever, and the views of Gassendi certainly deserve to be men- 
tioned. The important thing about Gassendi's natural philosophy 
in this connection was his admission of the vacuum as a primary 
conception. He supposed that atoms of light traversed the empty 
spaces between the celestial bodies and he left it open to ques- 
tion whether, by analogy with sound, these atoms were emitted 
periodically. Ideas obtained partly from Descartes, partly from 
Gassendi, seem to have been combined by Huygens but there 
was much that was new about his own theory. As has been men- 
tioned, Hooke, in his Micrographia of 1665, spoke in general 
terms of waves of light propagated with finite speed. Grimaldi, 
in his Phy$ico*nvtkesis de Lumine of the same year, pursued 
similar ideas; there is in this work a figure which may represent 
light as propagated by transverse vibrations. He spoke of the 
motion of a fluid medium as being spiral in form. Whether Huy- 



gens read this work is uncertain but it was amongst his books 
at the date of his death. 

Perhaps more importance should be attached to the work of 
the Jesuit I. G. Pardies (d. 1673), f r t ^^ s amateur student of 
science showed the greatest faith in the Analogy between light 
and sound. It is known that he showed his completed work to 
Huygens. The latter never accepted so close an analogy but one 
point of resemblance has been remarked: Pardies supposed that 
the pulses by which light was propagated were irregular in period. 
The work Optique, published in 1682 by Ango, contains some of 
his ideas. The first to develop the idea of periodicity in the pro- 
pagation of light was Newton (1672), and Malebranche was the 
first French writer to follow him (1699). Huygens seems to have 
been led to make an independent study of the problem as a 
result of the perplexing properties of Iceland spar; it is known 
that he conceived the idea of spheroidal waves within this crystal 
while at the Hague in 1677. There is little to be gained, however, 
by attempting to maintain that Huygens originated the wave 
theory de nova. He may well have obtained the initial idea from 
Pardies's manuscript. His great achievement lay in presenting the 
theory in a form in which it could be fruitfully applied through 
the development of a suitable geometry. 

Huygens did not invoke a Cartesian medium for the propa- 
gation of light until after 1668, when it seemed to him that he 
had definite evidence from his vacuum pump experiments for the 
existence of such a medium. He then sought to combine this 
hypothesis with an atomic theory similar to that propounded by 
Boyle. He agreed with the latter that liquids and solids alike are 
composed of particles in proximity but he supposed that there 
were intervening spaces and that these were filled with the much 
smaller particles of a subtle matter. The elasticity of the air 
demonstrated by Boyle's experiments "seems to prove," he wrote, 
that it is made up of particles floating and " agitated very rapidly 
in the ethereal matter composed of much smaller parts ". Huy- 
gens's atomism is peculiar therefore in that he extended it to 
subtle as well as to ponderable forms of matter. That a consider- 
able proportion even of solid bodies is occupied by the ethereal 
medium is clear, he considered, from the fact that so dense a sub- 
stance as gold does not screen off the effect of a magnet or of 
gravitation from a body. 

Huygens considered his kinetic theory of matter was only a 



beginning. He was forced to introduce " soft " particles to damp 
out the motion of light in opaque bodies and his speculations 
led him in one place to speak of the particles of ether being com- 
posed of still smaller parts and penetrated by a second subtle 
matter. There is some inconsistency in his statements and we are 
left in some doubt whether he always intended a clear distinction 
between a luminiferous ether on the one hand and subtle media 
for the propagation of gravitational and other effects on the 
other. He did distinguish an ether and a matiere subtile but the 
grounds of the distinction are not really clear. The nearest Huy- 
gens came to simplifying his ideas was to suggest that the media 
concerned in the propagation of light and in the anomalous 
vacuum pump experiments are the same. He explained that he 
could not identify the luminiferous medium with that which 
causes gravity since the latter was in his view found only near the 
earth. The weakness of his position here has already been dis- 
cussed. As Clerk Maxwell remarked, " To fill all space with a 
new medium whenever any new phenomenon is to be explained 
is by no means philosophical." 

The Traite 1 begins with an admission that some of the sug- 
gestions are only hypothetical : " whereas the Geometers prove 
their Propositions by fixed and incontestable Principles, here 
the Principles are verified by the conclusions to be drawn from 
them; the nature of these things not allowing of this being done 
otherwise. It is always possible to attain thereby to a degree of 
probability which very often is scarcely less than complete 
proof." It is in keeping with this view of scientific method 
to start with the hypothesis that light is a form of vibratory 
motion propagated in spherical waves or surfaces as in the pro- 
pagation of sound. The chief differences between the two cases 
were the incomparably greater speed of light estimated by Roe- 
mer and the media through which the vibrations travel. It was 
also obvious that the particles of a luminous body vibrate inde- 
pendently of each other and that the frequency of vibration (if 
one can use such a term in relation to Huygens's ideas) is very 
much higher than in the case of sound. Huygens then proceeded 
to explain how light could be conceived as a succession of com- 
pressional or longitudinal vibrations passing through contiguous 
ether particles. Contrary to the corpuscular theory there was 
no movement of translation. This explained how two light rays 
1 Traite de la Lumi&re. 


could travel in opposite directions in the same space, or cross at 
an angle, without hindering each other. It also led to Huygens's 
famous conception of secondary wavelets, for each particle in the 
path of a disturbance was a centre from which the disturbance 
spread outwards through all the particles in contact. Huygens 
saw that there was a difficulty here in that loss of impulses later- 
ally must weaken a ray as it proceeds. The limits he set to the 
" wave front " or common tangent appear quite arbitrary. As 
is well known, Newton was not satisfied with this discrepancy; 
it was only much later that wave-spreading in the form of diffrac- 
tion effects was fully demonstrated. 

Huygens's principle is often considered to be open to the 
objection that a wave-front travelling backwards towards the 
source can be constructed in theory but is never observed in 
practice. The answer to this objection is that Huygens did not 
evolve his principle from geometry so much as from his study of 
elastic collision. If the particles of a medium are all equal, any 
impulse received at A will be transmitted through the train of 
particles until any given particle C is moved in the same direc- 
tion as D. After collision with the next stationary particle E there 
will be a rebound only if C were smaller in mass than E. Huy- 

o ccooo o 


FIG. 48 

gens supposed all the particles of the ether to be of the same 
size but allowed that the effect would exist if smaller particles 
were present. He doubted if such a back-wave would generate 
the sensation of light. There is, however, something very un- 
satisfactory in considering a point of a wave as the centre of 
another wave. 

The construction for regular reflection at a plane surface is 
sufficiently well known to require only brief notice here. It is 
given in text books of optics. Instead of progressing towards 
GMMMMB the secondary wavelets can radiate outwards only 
above the reflecting surface AB. While a secondary wavelet from 
C is travelling to B a wavelet starting from A must have travelled 
through a distance AN equal to CB. The radii of intervening 
wavelets are determined in the same way and thus the common 


J G 

FIG. 49 

tangents BN can be found. This " terminates the movement " 
and is the new wave-front. By geometry it may readily be shown 
that, as a consequence of this construction, the angles of incid- 
ence and reflection are equal. 

The validity of Huygens's principle here depended, as its 
author saw, on the constancy of the speed of light during reflec- 
tion. Without naming the source he referred to Hooke's law in 
answering this question : " This [constancy of speed] comes 
about from the property of bodies which act as springs, of 
which we have spoken above; namely that whether compressed 
little or much they recoil in equal times/' 

It was otherwise when the passage of light from one medium 
to another was considered. An important difference and one 
which became decisive between the wave theory and the cor- 
puscular theory was Huygens's conclusion that light must travel 
more slowly in the denser medium. This was not necessarily be- 
cause the medium was different : Huygens considered the ethe- 
real medium penetrated all solids and liquids. The difference in 
speed resulted from the detours of the waves around the more 
solid particles of the elements. If we ask how Huygens knew 
that the ether penetrates substances his answer, given in the 
Traitf, adduces the entry of ether into the Torricellean space and 
that hollow bodies possess an inertia which is in strict proportion 
to the mass. Like Newton, he could detect no friction of solids 



with the ether. In some substances, however, Huygens supposed 
the material particles were not unaffected by the light vibrations. 
In this case they transmitted the vibrations also and the existence 
of this second mode explained double refraction. 

In his treatment of ordinary refraction, Huygens supposed a 
wave-front AC (Fig. 50) to impinge on the surface of a second 
medium. The speed of light in this second medium (glass) was 
supposed to be two-thirds of its value in air. The radius of the 
wavelet from A is then two-thirds of the distance which the 
wavelet would travel in air in the same time interval. It follows 
that while a wavelet from C is travelling to B, a wavelet from A 


FIG. 50 

will spread into the new medium through a distance AN equal to 
two-thirds of CB. It is clear in each case that the radius of the 
wavelet from K will be two-thirds of the distance KM which 
the wavelet would have travelled in air. The new wave-front is 
the common tangent NB. It follows by geometry that 

sin L DAE velocity in air 

sin L NA F ~~ ^ "" velocity in glass 

When light goes from glass into air the ratio of the speeds 
is inverted. In the second figure (Fig. 51), therefore, /!N=3/a BC 
or^l^AG. Here we have 

sin L DAE z 

sin L NAF = "3""" 
For larger angles of incidence (DAE) it is clear that when 

1 82 



sin LDAE = 

sin L NAF = 
sin Z DAE 




sin /.'NAF ~ 3" 

sin L NAF = i and the angle of refraction be- 
comes 90. Beyond this limiting condition, as Huygens puts it, 
the wave-front BN " cannot be found anywhere, neither conse- 
quently can A N . . . thus the incident ray DA does not pierce the 
surface AB ". 




FIG. 51 

It was impossible to give a satisfactory physical reason for 
the failure of the ray to penetrate the surface when the angle of 
incidence exceeded the critical value. The interior reflection 
which occurred took place, he supposed, " against the particles 
of the air or others mingled with the ethereal particles and larger 
than they ". He could not explain how the reflection could take 
place when the air was replaced by a vacuum. 

At the end of his chapter on refraction, Huygens showed that 
his principle of secondary wavelets was in conformity with Fer- 
mat's principle that the path actually taken by a ray of light 
in passing between two points is the path of least time. Fermat 
was in strong opposition to Descartes, whose false " demonstra- 
tion " of Snell's law of refraction required that the velocity of 
light be greater in a dense medium than in air. In a long letter 
to de la Chambre, Fermat explained that it was necessary to 
make the opposite assumption. He showed how the sine law 
of refraction could be deduced from the least-time principle and 
a copy of the letter was sent to Huygens in 1662. Huygens 


was at first scornful of Fermat's principle, which seemed to him 
to savour of Aristotelianism. This " pitiahle axiom " was one 
which he had never seen usefully applied, he remarked. Never- 
theless, he changed his mind over Fermat's principle and was de- 
lighted when he succeeded in deriving it by his own methods. 

The relation between Fermat's principle and Huygens's con- 
struction in this case may be summed up by the statement that 
both give the same physical interpretation of the refractive 
index. The significance of this important theorem was not lost 
on Huygens, though it is to be doubted if he ever imagined its 
future development. There is, however, an interesting applica- 
tion of the least-time principle in the Traite to atmospheric 
refraction, a subject of obvious importance to astronomers. 
Huygens pointed out that spherical wavelets would only be set 
up in a medium which was homogeneous, or to use the modern 
term, isotropic. In an anisotropic medium, as will be seen, 
Huygens showed that the wavelets might be ellipsoidal in form. 
The problem of the atmosphere was somewhat different. There 
was here a gradual change of refractive index with density, and 
Huygens saw that the wavelets must have surfaces of equal time 
from the source. To include the effect of density in his wave 
theory it was necessary to assume either that gaseous particles 
acted as a hindrance to the vibrations or else that they trans- 
mitted light themselves but did this inefficiently. 

Huygens himself foresaw that later workers would furnish 
what was needed to complete his " imperfect knowledge/' that 
much yet remained to be done to make a satisfactory theory. 
The remainder of the Traite was taken up with a striking 
attempt to extend the wave theory to the phenomenon of double 
refraction. As is well known, Huygens elucidated the nature of 
the wave surface for the extraordinary ray as an ellipsoid of 
revolution and this great achievement still stands. What 
Huygens could not explain (since his " waves " were longitu- 
dinal ones) was the effect of superimposing two calcite crystals in 
different positions. He succeeded, that is, in working out a con- 
struction for the extraordinary ray on the basis of an ellipsoidal 
wave theory, but could not on these grounds account for what we 
now term the polarization of the transmitted light. 

The properties of calcite are described in some of the larger 
works on optics, notably Mach's The Principles 'of Physical 
Optics. The essential property is that, in general, an incident 

i8 4 


ray, on entering the crystal, is split up into two rays, one of 
which (the ordinary ray) is refracted according to the usual laws 
of refraction, while the other (extraordinary ray) is not. The 
substance was, Huygens admitted, anomalous and its behaviour 
" seemed to overturn our preceding explanation of regular re- 
fraction ". From the first, however, Huygens, on account of his 
atomic theory, inclined to the view that the optical properties 
of crystalline solids were to be correlated with their fine structure. 
He therefore began his account in the Traite with a description 
of the geometry of the Iceland spar crystal. 

The crystal (Fig. 52) has the form of an oblique parallel- 
epiped and there are cleavage planes in three directions parallel 
with the three pairs of parallel faces. The angles of the parallelo- 
gram sides were given by Huygens as 101 52' and 78 8'. The 
crystal has two opposite corners which are formed by three 
obtuse angles while the others are formed each by two acute and 
one obtuse angles. If the obtuse angle ACB of the parallelo- 
gram face at the blunt corner C is bisected by the line CE and a 
plane is imagined to pass through CE perpendicular to the 

D E 

FIG. 52 

parallelogram face, the plane also contains the edge CF. The 
plane thus determined, and any other plane parallel to it, was 
termed by Huygens a principal section. 

As Bartholinus had shown, every incident ray, with certain 
exceptions, gave rise to two refracted rays, one of which 
was normal in its behaviour and exactly comparable with the 
refracted ray in any ordinary medium. The second ray showed, 
in general, marked abnormalities. So long as the plane of incid- 
ence coincides with a principal section HH (Fig. 53), bath of the 


refracted rays remain in this plane. For other planes of in- 
cidence the extraordinary ray is formed in a different plane. 
Moreover, while a ray S incident normally and an oblique ray 
R making a certain angle of incidence, both in the plane of a 
principal section, are refracted in the normal manner (rays 
marked O), the extraordinary ray formed exhibits a peculiarity. 
This is that a ray incident normally in the principal section 
gives an extraordinary ray e which is deviated by 6 40' towards 
the blunt corner C; on the other hand, a ray R, incident in the 
principal section at 73 20' (almost parallel with the edge CF in 
the first figure) gives an undeviated extraordinary ray e. 

By a method which was in effect the measurement of the 
true and apparent depth of a small object seen through the 
crystal, Huygens found that the refractive index for ordinary 
rays in the principal section (or any other plane) was constant 
and approximately 5/3. Using a similar procedure to find 
the refractive index for extraordinary rays it was obvious that 
this was not a constant. The apparent depth of a point source of 
light varied with the orientation of the crystal. Nevertheless, 
Huygens discovered one important rule for the extraordinary 
refraction which may be explained as follows. In Fig. 54, the 
parallelogram GCFH is the principal section. 

IK is a ray normal to the surface and KM is the extraordinary 
ray. Huygens found that for rays VK, SK, making equal angles 
on either side of the normal IK, the extraordinary refracted rays 
KX and KT make MX and MT in HF equal. This may be called 
Huygens's rule for the extraordinary refraction in this plane. 



These facts had, if possible, to be collated in a single theory 
of transmission. The ordinary refraction offered no difficulties. 
For this the theory of spherical wavelets spreading with a speed 
less than that in air was adequate. " As to the other emanation 
which should produce the irregular refraction," Huygens wrote, 
" I wished to try what elliptical waves, or rather spheroidal 
waves would do ... ". These perhaps spread indifferently in 
both ethereal and material particles and in the regular arrange- 
ment of the particles might lie the source of the spheroidal wave 
form : " I scarcely doubted that there were in this crystal such an 
arrangement of equal and similar particles because of its figure 
and of its angles with their determinate and invariable 
measure ". 

Working on this assumption, Huygens's construction for the 
extraordinary ray from normal incident light was as follows. In 
Fig. 55, RC is a wave front and AB is the surface of the crystal. 
The plane is that of the principal section. Hemispheroidal 
waves originate at AKkkB. The axes or major diameters of 
these are oblique to the plane of AB as shown by AV: "I say 
axis or major diameter because the same ellipse SVT may be 
considered as the section of a spheroid of which the axis is AZ 
perpendicular to AV" wrote Huygens. For the present he 
considers only sections of the spheroid which are elliptical in the 
given plane of the figure. The common tangent to the semi- 
ellipses is NQ; and this is the propagation of RC as in Huygens's 
original theory. NQ is parallel with AB, but is displaced later- 
ally as required by the refraction of the extraordinary ray. 

R H h h C 

Fic - 55 


It was next necessary to find the exact form of the ellipsoid of 
revolution and the orientation of axes in the crystal l . Fortun- 
ately all six faces of the parallelepiped produce the same refrac- 
tions : the substance was uniaxial. Picturing the appearance of 
the blunt corner of a calcite rhomb (Fig. 56), and imagining the 
three principal sections respectively normal to each of the three 
faces, these intersect in a line, called by Huygens the axis of the 
corner, subtending equal angles with each of the three edges to 
the corner. If, now, the direction of the axis of the wave 
spheroid of rotation of which Huygens first thought did not 
coincide with that of the axis of the corner, each of the three 
principal sections would not be characterized by the same optical 

FIG. 56 FIG. 57 

properties. The inclination of the axis of the corner to each of 
the faces of the corner amounts to 45 20'. The orientation of 
the spheroid being known, the fact that for normal incidence the 
extraordinary ray is deviated in the principal section from the 
ordinary ray by 6 40' towards the blunt corner is sufficient to 
establish the shape of the spheroid. By calculations based on his 
data, Huygens found the following to be in agreement with the 
facts. If OA (in Fig. 57) is the axial direction of the calcite and 
the ordinary wave spreading out from the point O of the crystal 
is represented by a sphere of radius OA, the surrounding oblate 
spheroid of rotation AB with axis of rotation OA represents the 
corresponding extraordinary wave emerging simultaneously 
from O. The ratio of OA to OB is as 8 to 9 (very nearly) while the 
ratio of OA to the corresponding path in air is as 3 to 5. 

The construction for the refracted rays from oblique incident 
light may be explained as follows. The plane of incidence is con- 

1 I am indebted for this passage to the excellent summary of Huygens's 
Ttaitd in Mach's Principle* of Physical Optics. 


sidered to lie in the plane of the paper; MN is the calcite-air sur- 
face (Fig. 58) and SO an incident ray. 

Let SO bfc produced to S' and let a sphere of any convenient 
radius be described about O as centre. At its intersection with 
SOS 7 let a tangent plane be constructed intersecting MN in Q. If 
now a sphere of three-fifths the radius of the former is described 

M \0 Q N 


about O, its point of contact R with the tangent plane through 
Q gives the ordinary ray OR. Let OA be the direction of the 
axis of the calcite rhomb. A spheroid is now described about the 
smaller sphere such that its axis of rotation OA (the minor 
axis), which equals OR, is 8/9 the length of the major axis. 
The point of contact T of the tangent plane through Q to the 
spheroid then gives the extraordinary ray OT. This construction 
is confined to one plane only when the axis is symmetrical with 
respect to the plane of incidence, that is, either coincident with 
it or perpendicular to it. For any other orientation the extra- 
ordinary ray is inclined to the plane of incidence. The results 
were studied experimentally by Huygens, who cut the crystal 
so that the optic axis was normal to the surface, parallel with the 
surface and in the plane of incidence, among other forms. He 
also found experimentally that for rays in the principal section 
the extraordinary ray was not refracted when the angle of in- 
cidence was 1 6 40'. What he had to do here was to show that 
for this angle the rays continue without refraction since they 
are directed along the major axis of the ellipsoid. The fact that 
there is no bending in spite of the change in velocity in the new 
medium is of course explained by the fact that the new wave 


front is not normal to the direction of the rays as is the case for 
spherical wavelets. 

On the basis of his theory, Huygens clearly expected that it 
would in general be possible to split up rays which emerged from 
one crystal of calcite by passing them through a second crystal. 
Excluding certain special positions of the crystals which he him- 
self understood, it would be expected that the ordinary and 
extraordinary rays would be split up again on entering a second 
crystal. This was found to be by no means the case. Huygens 
was considerably perturbed by the discovery and laboured hard 
to explain it, but in vain. He was, as Mach remarks, on the 
threshold of a great discovery the transverse nature of light 
waves but his conceptions hindered his taking this step for- 
ward. "Before finishing the 'treatise on this crystal/' he wrote, 
" I will add one more marvellous phenomenon which I dis- 
covered after having written all the foregoing. For though I have 
not been able till now to find its cause, I do not for that reason 
wish to desist from describing it, in order to give opportunity to 
others to investigate it. It seems that it will be necessary to make 
still further suppositions besides those which I have made; but 
these will for all that not cease to keep their probability after 

\A /A 

having been confirmed by so many tests." In this Huygens was 
correct, his geometrical analysis of the ellipsoidal wave still 
stands. He went on (Fig. 59) : " The phenomenon is, that taking 
two pieces of this crystal and applying them one over the other, 
or rather holding them with a space between the two, if all the 
sides of one are parallel to those of the other, then a ray of light, 


such as AB, is divided into two in the first piece, namely, BD 
and BC, following the two refractions, regular and irregular. On 
penetrating thence into the other piece each ray will pass there 
without further dividing itself in two; but that one which under- 
went the regular refraction, as here DG, will undergo again only 
a regular refraction at GH; and the other, CE, an irregular re- 
fraction at EF. And the same thing occurs not only in this dis- 
position, but also in all those cases in which the principal section 
of each of the pieces is situated in one and the same plane, with- 
out it being needful for the two neighbouring surfaces to be 

In these words Huygens described his discovery of the 
polarization of light. He went on: " Now it is marvellous why 
the rays CE and DG, incident from the air on the lower crystal, 
do not divide themselves the same as the first ray AB. One would 
say that it must be that the ray DG in passing through the upper 
piece has lost something which is necessary to move the matter 
which serves for irregular refraction." "... It seems that one is 
obliged to conclude that the waves of light, after having passed 
through the first crystal, acquired a certain form or disposition in 
virtue of which, when meeting the texture of the second crystal, 
in certain positions, they can move the two different kinds of 
matter which serve for the two species of refraction; and when 
meeting the second crystal in another position are able to move 
only one of these kinds of matter. But to tell how this occurs, I 
have hitherto found nothing which satisfies me/' The twenty- 
sixth query at the end of Newton's Optics referred to this 
problem. Has not a ray of light two sides, Newton asked, and 
his question became one of extreme significance after Young 
made the suggestion that the wave motion of light is not 
longitudinal but transverse. 

Huygens did not in the Traite de la Lumierc attempt a 
detailed physical explanation of the production of a spheroidal 
wave but he communicated on this subject with Papin. In a 
letter written in December 1690, he wrote: "As to the kinds of 
matter contained in Iceland crystal, I suppose one composed of 
small spheroids, and another which occupies the interstices 
around these spheroids, and which serves to bind them together. 
Besides these, there is the matter of ether permeating all the 
crystal, both between and within the parcels of the two kinds 
of matter just mentioned; for I suppose both the little spheroids, 


and the matter which occupies the intervals around them, to be 
composed of small fixed particles, amongst which are diffused 
in perpetual motion the still finer particles of ether. There is 
now no reason why the ordinary ray in the crystal should not be 
due to waves propagated in this ethereal matter. To account for 
the extraordinary refraction, I conceive another kind of waves 
which have for vehicle both the ethereal matter and the two 
other kinds of matter constituting the crystal. Of these latter, I 
suppose that the matter of the small spheroids transmits the 
waves a little more quickly than the ethereal matter, while that 
around the spheroids transmits these waves a little more slowly 
than the same ethereal matter . . . These same waves, when 
they travel in the direction of the breadth of the spheroids, meet 
with more of the matter of the spheroids, or at least pass with 
less obstruction, and so are propagated a little more quickly in 
this sense than in the other; thus the light-disturbance is propa- 
gated as a spheroidal sheet/' Papin and Huygens did not, how- 
ever, see eye to eye over atomic doctrines and Huygens was 
critical of Papin's more or less orthodox Cartesian ideas. 

Professor Sylvanus Thompson, in his " Note by the Trans- 
lator " which prefaces the English edition of Huygens 's Traite, 
remarked : " If Huygens had no conception of transverse vibra- 
tions, of the principle of interference, or of the existence of the 
ordered sequence of waves in trains, he nevertheless attained to a 
remarkably clear understanding of the principles of wave- 
propagation; and his exposition of the subject marks an epoch in 
the treatment of optical problems." There seems to be fairly 
general agreement that Huygens's theory ought more properly 
to be called a pulse theory. The concept of periodicity was no- 
where introduced. Some evidence, though not of very substan- 
tial character, can be found in Huygens's note-books to show 
that he did speculate that waves (ondes) exist. He did not show 
much interest in the problem of accounting for colour, and 
although he made some favourable comments on Hooke's ex- 
planation of colour, he himself felt the matter lay outside mathe- 
matical treatment. Newton first introduced the idea that colour 
is related with frequency, but he received scant encouragement 
from Huygens. It is curious that Newton, the founder of the 
corpuscular theory, at least in his later speculations, came nearer 
to a wave theory of the modern type than Huygens did. He 
never threw over the corpusular theory, however, on the grounds 


that first, a wave motion should show some lateral spreading (not 
yet identified with Grimaldi's discovery of diffraction) and 
second, that no waves as then conceived could produce the effects 
discovered by Huygens when light passed successively through 
two crystals of Iceland spar. As Whittaker remarked, " his 
objections are perfectly valid against the wave theory as it was 
understood by his contemporaries, though not against the theory 
which was put forward a century later by Young and Fresnel ". 
The idea of transverse vibrations was, in Whittaker's opinion, 
narrowly missed by John Bernoulli the younger, whose prize 
essay on the ether was written in 1736. 

The very last part of Huygens's Traitt is an echo of 
Descartes. Here Huygens wrote on " the Figures of the trans- 
parent Bodies which serve for Refraction and Reflection ". 
Here Huygens treated problems of aplanatic surfaces by means 
of his wave construction and by employing the least-time prin- 
ciple. In this way he gave proofs for some of the propositions 
Descartes had merely stated and for which his method had not 
been recorded. Huygens acknowledged the importance of conic 
sections. He proved that concavo-convex lenses of the form 
described by Descartes would be aplanatic, and determined the 
conjugate foci. Following Descartes he proposed that the concave 
surface of such a lens should be spherical. The impression given 
by the work, however, is that its interest is from the start purely 
theoretical. Huygens certainly recognized that it was of little 
use to demonstrate that refracting surfaces of unlimited aperture 
could be aplanatic so long as chromatic aberration was without 
remedy- His constructions for reflection from curved surfaces 
were, however, of permanent value and his general method of 
finding the focus from a consideration of wave fronts has proved 
most useful. 

There is no question that Huygens's principle of the propa- 
gation of light by secondary wavelets has been of great import- 
ance in optics. Many of the questions left unanswered by 
Huygens were settled by Fresnel. But the early history of 
Huygens's wave theory was disappointing. It is true that the 
Traite de la Lumtere was well received. The wave theory was 
even expounded at the University of Wittenberg in 1693. Soon 
after Huygens's death, however, the theory seems to have fallen 
into general neglect. In Holland neither s'Gravesande nor 
Musschfenbroek adopted it. 



As has been described in the notes on Huygens's life, the 
planet Saturn presented a puzzling appearance when viewed 
through the first telescopes. Jupiter's satellites were a spectacle 
which could not but strengthen belief in the principles enunci- 
ated by Copernicus, but this was far from being the case with 
Saturn. Galileo ended his studies of Saturn with the words 
altissimum planetam tergeminum observavi, but concealed in the 
form of an anagram : " I have observed the most distant planet 
to be triform." A system of three bodies, two of them rotating 
about the middle one would, however, be a sufficiently curious 
spectacle to arouse further attention. 

Some account of the contents of Huygens's Systema has been 
given in the early part of this book. The work of observing the 
planet, when the weather was favourable, occupied Huygens 
during the years 1655-9, and this involved a considerable 
amount of work in the construction of telescopes. The rings 
were fortunately becoming more inclined during this period, and 
in 1664, five years after his book had appeared, Huygens drew the 
rings at a great inclination. He was, of course, mistaken in sup 
posing a single solid ring. The Cassini division is visible in a small 
telescope only when the rings appear fully open, a condition 
which was not fulfilled during Huygens's earlier studies. In 1675, 
however, he made a drawing which suggests that he abandoned 
the view that the ring is solid (page 194). 

In regard to the sidereal period of the planet and the constant 
inclination of the rings to the ecliptic Huygens was more or less 
correct. He remarked in his preface on the equal inclination of 
the axes of Saturn and the Earth to the ecliptic, and he made 
it abundantly clear that the book was essentially a comment on 
Copernicanism. He accordingly did not consider it out of place 
to mention his search for moons of Mars, Venus and Mercury. 
In observing the four known moons of Jupiter, he remarked a 
variation in the appearance of the disc of the planet which he 
ascribed to the presence of vapours comparable with our clouds. 
Less related with his main subject, but worthy of record, was his 
N 193 



I. p. 322. One of the 
first drawings. 

II. p. 224. Letter of Sept 
24, 1658. 

V. p. 109. In letter to 
Moray, Aug. 29, 1664. 

VI. p. 443. Saturn 
observed May, 1669 
(Huygens, Picard and 

A drawing from 
Huygens's Manuscript 
K. (1675). 

Dec. 27, 1657. 

Drawings of Saturn after those made by Huygens 
The references are to the volumes of the Oeuvres Computes. 


description of a nebula in Orion. This nebula, discovered by 
Cysat in 1619, he considered to be essentially different from the 
Milky Way, in that it did not lose its nebulosity when viewed 
through a telescope. He suggested that the light came from a 
more distant region visible through a hole in the black sky. 
Much later (1733), Derham questioned if the nebulae "in all 
Probability, be Chasms, or Openings into an immense Region of 
Light, beyond the Fix'd Stars." 

In regard to Saturn, Huygens's method was to collect together 
all drawings of the planet then published and show that nearly all 
could be explained on the hypothesis of an imperfectly visible 
ring. From the outset, as he himself admitted, it had always 
seemed obvious that the planets were analogous with the Earth; 
consequently it is probable that all turn on their axes. A rather 
more dangerous proposition was that all the matter between 
Saturn and its moon (Titan) turns about the planet with periods 
increasing with the radius. But it seemed obvious that the dis- 
position of rotating matter was at any rate symmetrical, for the 
appearance of the planet changed only slowly. Huygens found 
that the plane of the ring was inclined at about ao to the 
ecliptic and that this angle remained constant as in the case of 
the inclination of the Earth's equator. He could not accept any 
suggestion that the ring was merely an exhalation and an 
evanescent phenomenon. The solid and permanent nature of 
the ring could, he said, be clearly perceived. The figure he gave 
in the Systema (Fig. 60) left no room for doubt as to his own 
view on this point. But such a ring was certain to appear 

FIG. 60 

eminently unstable unless one accepted its equilibrium under 
central gravitational forces. This Huygens proposed was the 
true explanation. He did not suggest that the gravitational force 
on the elements of the ring was balanced by the centrifugal 
force due to rotation and thus did not anticipate Newton's 


theory. His conception was a static one and required that the 
ring be mechanically able to withstand the gravitational effect 
without fracture. Rotation then did not come into the question. 
But the idea that Saturn's gravity extended to the ring may well 
have been an important advance, and may even have started 
Newton on some interesting speculations. 

Having disposed of effects due to aberration and obscurity of 
early telescopes, Huygens was faced with the necessity of show- 
ing how the hypothesis of a ring inclined to the ecliptic pro- 
vided an explanation of all the various appearances of the 
planet. With the telescopes used at the time the ring appeared 
invisible when there was anything up to 2 between the line of 
sight and the plane of the ring. The problem of forecasting the 
future appearances was a purely geometrical one. In Fig. 61, 
ANC represents the orbit of Saturn, DEF that of the Earth, and 
L the position of the Sun. The orbits are regarded as circular 
and lie in the same plane. Since the inclinations of the axes of 

FIG. 6 1 

the Earth and of Saturn are parallel, the equinoctial line AC 
passes through the equinoctial points for both planets. Now 
since the axis of Saturn remains parallel with itself the line 
of intersection of the planes of the ring and the orbit is always 
parallel with the line AC. If Saturn is at H, and the Earth at D, 
and HM is the line of intersection of the planes of the ring and 
orbit, if the angular displacement of the Earth from the line AC 


is greater than that of Saturn, the line HM will fall between the 
Sun and the Earth. The plane of the ring passes between L and 
D and no reflected light from the ring can reach D. On the 
contrary, when the Earth is at less angular displacement than 
Saturn, as in the relative positions N and F, or on opposite sides 
of CA, as at N and f, the same surface of the ring would be 
visible from either F or f. 

Knowing the synodic period of Saturn, that is the interval 
between two successive oppositions of the planet, it was possible 
to calculate the dates of reappearance of the round form. 
Huygens showed that the ring would appear thin from April to 
June 1671, and vanish from sight in July or August. Not until 
July or August 1672 (after an heliacal rising and setting) would 
the ring appear again. It would then remain visible until 1685. 
In this year, and again in 1 700, the planet would appear in the 
round form. Actually, Huygens had to recognize errors in these 
predictions since events were slightly ahead of schedule even in 
1671. Huygens did not make very exhaustive observations, and 
it appears likely that his fundamental determinations were not 
sufficiently refined. He did not attach great importance to long- 
continued and uninterrupted observation. Indeed, many of his 
observations passed unrecorded. 

The telescope first used in the observation of Saturn had a 
magnification of about fifty. The objective was plano-convex, 
and the eyepiece was a simple lens of about eight centimetres 
focal length. An objective answering to Huygens's description 
of it, and dated February 3rd, 1655, was discovered at Utrecht 
University in 1867 by Harting (who wrote a short biography of 
Huygens in Dutch). Huygens's second telescope was one of 23 
feet about twice as long and twice as powerful as the first. 
This was used after February 19, 1656. The method of com- 
puting the magnification was either by determination of the 
ratio of the focal lengths of the objective and the eyepiece or by 
comparison of the angular dimensions of a distant object seen 
through the telescope and by direct vision. 

In his work on Saturn, Huygens used a micrometer consisting 
of a lamina of brass. This form, which has already been men- 
tioned, was used up to the end of 1659. In 1666, he adopted the 
use of cross wires arranged in squares. The movable thread 
micrometer invented by Gascoigne superseded this. The values 
for the diameters measured by Huygens were all too large, but 


they were an improvement on those given by Riccioli. Also 
they bear a very fair relation among themselves. As is shown 
in the table, Huygens's ratios foil the diameters of the planets 
compared with the Sun are all in the direction of over-estimating 
the planetary diameters. 

Planet. Ratio of diameter to True values, 

that of Sun. 

Venus i 

Mars i 

Jupiter i 

Saturn's ring i 

84 i : 112 

1 66 i : 202 

5-5 ' : 9-8 

7.4 i : 1 1.6 

While studying the apparent diameters of Jupiter and Mars, 
Huygens noted the existence of bands or zones across these 
planets. Drawings were given in the Systema Saturnium. 

Huygens adopted Copernicus's proportions for the planetary 
distances from the sun. There was, until 1672, no agreement 
among astronomers over the distance of the earth from the sun. 
It was therefore necessary to proceed on a probable estimate if 
such could be found. Huygens's method was, having observed 
the apparent diameters given above, to adopt for the Earth a size 
which agreed best " with the order and good disposition of the 
whole system ". In a way which reminds one of Kepler, this dis- 
position seemed to Huygens to rest on a proportionality between 
size and distance from the sun (Jupiter and Saturn being excep- 
tional). Thus the earth, being intermediate between Mars and 
Venus, probably possessed a volume intermediate between these 
planets. From the figures given above, the mean (of 1/166 and 
1 784) is i / 1 1 1 . Huygens took the estimate that the sun's diameter 
was i / 1 13 of its mean distance from the earth. This gave the dia- 
meter of the earth as i / 12543 of the sun's mean distance. The 
maximum and minimum distances of Saturn came out at 
122000 and 100344 terrestrial diameters respectively. A modern 
estimate would put the result at about 123600 and 100200 respect- 
ively, so that, considering the precarious basis of Huygens's 
calculation, the result was much better than might have been 

In his account of his method of observing apparent diameters 
Huygens explained that he used a diaphragm at the focus of the 
objective. This diaphragm had a hole a little smaller than the 
diameter of the eye-piece and in this way a sharp edge to the field 


of view was obtained. It was easy to find the angular size of 
the field of view by timing the passage of a star across it, using 
the pendulum clock " recently invented ". His own telescope (pre- 
sumably the one of twenty-three feet) embraced a field of o if 
15". The description of a micrometer consisting of copper rods 
of diminishing diameters which could be inserted in the focal 
plane of the eye-piece ends the Sy sterna. 

Huygens made further observations of Saturn in later years 
and redetermined, among other things, the ratio of the dia- 
meters of the ring and the planet. On July 16, 1667, Huygens 
determined, probably with Buot, the hour at which the ring ap- 
peared parallel with the horizon at Paris. This enabled him to 
calculate the inclination of the ring to the ecliptic, but the details 
of the work are lacking. The most important work on Saturn in 
later years was done by Cassini, working at the Paris observatory, 
Cassini discovered a second satellite (October 1671) and observed 
the division in the ring which now bears his name. 


During the later part of the seventeenth century, Fontenelle, 
historian of the Academic Royale des Sciences, attracted much 
attention by writing descriptions of the earth as seen by hypo- 
thetical inhabitants of Mars and Saturn. The little work, Cosmo- 
theoros, written by Huygcns, and published posthumously in 
1698, appears therefore to have been written in imitation of Fon- 
teneJle and its chief interest is that it shows Huygens, at the 
time of its composition, as notably Cartesian in outlook. It also 
shows him, as Leibnitz had hoped, in a more human light than 
most of his published work. The work was translated into 
French by Dufour in 1702 and into English in 1722 when it 
cnme out under the title The Celestial Worlds Discovr*d; it 
is from this work that quotations have been taken. 

Huygens began by remarking that scientific conjectures about 
the planets should not be judged contrary to the scriptures nor 
useless or impious. On the contrary, " besides the Nobleness and 
Pleasure of the Studies, may not we be so bold as to say, they 
are no small help to the Advancement of Wisdom and Morality, 
so far arc they from being of no use at all? For here we may 
count from this dull Earth, and viewing it from on high, con- 
sider whether Nature has laid out all her Cost and Finery upon 
this small Speck of Dirt." The English style of the translation 
takes one far from the spirit of Huygens's Latin. 

Nevertheless the work is throughout in an intimate vein, 
being written in the form of a letter to Huygens's brother Con- 
stantin. When Huygens outlined the Copernican theory he gave 
a diagram " like what you have seen in my Clock at home ". Per- 
haps he was referring to his machine for showing the planetary 
motions. At all events, the reader recognizes much of Huygens's 
early work in new dress the planetary magnitudes, the planets' 
distances from the earth and his views on the earth itself as a 
planet. "... we are so skilful nowadays," runs the translation, 
" as to be able to tell how much more or less the Gravitation in 
Jupiter or Saturn is than here . . . ". His general view was that 
the planets are so like the earth in most essentials as to have 



inhabitants of some kind in all probability. Among these essen- 
tials he reckoned the existence of water but perhaps with differ- 
ent properties from our own. It must have a lower freezing point 
on the cold planets. If some kind of human life exists, he sug- 
gested, there must be other forms of life upon which the human 
beings would be dependent. Man, he thought, on these other 
planets, probably had the same vices and the same power of rea- 
son. If his senses were not too different his studies were prob- 
ably the same as ours. Huygens pictured the night sky as it would 
appear to the inhabitant of Jupiter or Saturn. Letting his imagin- 
ation go, he remarked: " What a wonderful and amazing Scheme 
have we here of the magnificent Vastness of the Universe! So 
many Suns, so many Earths, and every one of them stock 'd with 
so many Herbs, Trees and Animals, and adorn 'd with so many 
Seas and Mountains! And how must our Wonder and Admira- 
tion be encreased when we consider the prodigious Distance and 
Multitude of the Stars! " " I must be of the same Opinion with 
all the greatest Philosophers of our Age," he added, "that the 
Sun is of the same Nature with the fix'd Stars." He criticized 
Kepler, therefore, for making the Sun superior to all other bodies 
in the sky. 

At the end he stated his modified vortex theory. " I am of 
the Opinion," he wrote, " that every Sun is surrounded with a 
Whirl-pool or Vortex of Matter in a very swift Motion; tho' not 
in the least like Cartes'* either in their Bulk or manner of 
Motion. For Cartes makes his so large, as every one of them to 
touch all the others round them, in a flat Surface, just as you have 
seen the Bladders that Boys blow up in Soap-suds do; and would 
have the whole Vortex to move round the same way/' Descartes's 
views, he asserted, needed to be corrected in the light of New- 
ton's work, in particular to take account of the gravity of the 
planets towards the Sun and how " from that Cause proceeds the 
Ellipticity of the Orbs of the Planets, found out by Kepler ". 
His own vortices, he explained, were composed of matter 
which does not move all in the same way, "but after such a 
manner as to have its Parts carry 'd different ways on all Sides. 
And yet there is no fear of its being destroyed by such an 
irregular Motion, because the ^Ether round it, which is at rest, 
keeps the Parts of it from flying out." 

Such were Huygens's last words on the solar system. To the 
modern reader they seem curiously conflicting views. Huygens 


accepted all the mathematical part of Newton's work hut not his 
interpretations. He was unable to accept a purely empirical view 
of gravity and, as we have seen, considered his own experiments 
on the existence of an ether were decisive. In his own way, Huy- 
gens was only seeking to banish what he conceived to be occult 
properties. " Le grand mrite de Descartes est avoir vu que le 
probleme du monde est un probteme de mecanique," wrote 
D'Alembert. This outlook was certainly the one which was con- 
sistently adopted by Huygens. He did not see that his own work 
was in far better accordance with the Newtonian system, that 
it helped to expose and did not heal the wounds inflicted on Des- 
cartes 's natural philosophy. 

The Cosmotheoros reveals Huygens's religious outlook to a 
far greater extent than his other published works. It does not 
appear that he was ever a proclaimed rationalist, and indeed the 
evidence rather shows, on the contrary, that he continued to 
support Protestantism up to the end of his life. There were many 
points, however, on which his beliefs appear to have been 
unorthodox. While, for example, he considered that the consti- 
tution of the world argued the existence of an intelligent power 
behind phenomena, he felt that whatever divinity belonged to 
man belonged to his rational mind; through this mind man can 
apprehend the ways of the Creator, but equally this mind pre- 
cluded an acceptance of the cruder superstitions. Huygens does 
not seem to have accepted belief in the Devil, and perhaps he 
rejected personal immortality. Yet his outlook clearly belongs 
to that noble period of Protestant thought which found nothing 
alien in the new and enlarged horizons revealed by Science. As 
an admirer of the writings of Cicero, Huygens refers to the glory 
to be found in Nature as in some way a warranty that man is not 
without his significance in the scheme of things. Stoic philosophy 
found a response within the breast of this seventeenth century 
scientist, and one is reminded that from the beginning of the 
Christian era there had been an infiltration from this philosophy 
of the grand belief in the ultimate rationality of the world. This 
belief Huygens certainly supported; it was, one might say, the 
essence of his religious outlook. 

The Place of Huygens in the History of Science 

Descartes and Galileo were the brightest stars in the scientific 
firmament at Huygens's birth and they influenced him one way 
or another all his life. In Huygens's early days the world of 
science may be said to have been divided into those who followed 
the empiricism of Galileo and those who, with Descartes, 
ultimately distrusted it. This division leaves on one side, however, 
the Aristotelians who, mainly for religious reasons, could not 
accept Descartes's ingenious reconciliation of Copernicanism 
with dogma. The universities which, on the whole, sheltered the 
Aristotelians were in consequence left behind by the faster-mov- 
ing currents of contemporary thought. In time, it is true, they 
became affected by Descartes's thought and this acted as an 
introduction for more scientific ideas. Newton's views were first 
introduced at Cambridge as a sort of commentary on Descartes; 
at Oxford, the Sjivilian professorship of Astronomy almost alone 
was renowned for progressive ideas. The scientific societies of 
the early part of the seventeenth century were unacademic, 
amateur, spontaneous offshoots of the artistic renaissance, owing 
their existence partly to the rediscovery of Greek writings of 
analytical character, partly to the conflict of ideas in Astronomy 
and the development of Mathematics, and partly to the writings 
of Francis Bacon and of Descartes. The sources of the scientific 
renaissance include other tributary streams but when all is said 
the explanations seem incomplete and hardly concern us here. 
The important fact is that Huygens was born in the period when 
these scientific societies were in their infancy. As a young man, 
he must have heard of the work of the Florentine Accademia del 
Cimento and of the interest kindled in Paris through the reports 
of the well-known traveller Pieresc, who visited Florence. 
Huygens early studied the works of Galileo and through 
Mersenne obtained his introduction to a small world which was 
preoccupied with new problems and becoming more and more 
confident that it possessed a new technique of discovery. 

For the attack on Aristotle's science had been long prepared. 
Roger Bacon, Da Vinci, Benedetti and Stevinus preceded Galileo 



and began the initial liberation of Mechanics. The first criti- 
cisms, however, dealt with certain of Aristotle's postulates rather 
than with his entire method and Galileo was the first to replace 
the substance and attributes of scholastic description by what 
we must term scientific data. The whole movement of the seven- 
teenth century empiricists, then, took this direction. Instead of 
substance, essence, matter, form and other categories adapted to 
Aristotle's logic, an analysis was developed using those of 
space, time, mass, force and the like. The categories of 
thought were transformed. All this Huygens himself realized 
remarkably clearly. He perceived that even the sixteenth century 
writers had retained many of the occult properties of the Aris- 
totelians. Gilbert, Telesius and Campanella, he noted, had not 
enough inventiveness or mathematics. Even Gassendi was not 
much better. Bacon had seen the insufficiences of Aristotle and 
had in addition pointed out good methods for building a better 
system, but, he wrote, " he did nothing to advance mathematics 
and lacked penetration for physical matters, not having been 
able to conceive the motion of the Earth, at which he mocked *'. 
" Galileo, on the other hand, had the mind and all the knowledge 
of mathematics he needed to make progress in Physics and it 
must be admitted that he had been the first to make fine dis- 
coveries touching the nature of movement, although he left very 
considerable parts of it to be done. He had not sufficient boldness 
nor presumption to wish to undertake the explanation of all 
natural causes, nor the vanity to wish to be chef de secte. He was 
modest and loved truth too much; he believed besides that he had 
acquired enough fame and that, through his new discoveries, it 
would last for ever." 

Another feature of contemporary scientific thought was the 
assumption of a mathematical simplicity in the relations trace- 
able between data. This assumption is found in the work of 
Galileo and also in that of Kepler, Copernicus and, to some ex- 
tent, perhaps, even in that of Da Vinci. Kepler expressed the 
idea in two aphorisms : " Natura simplicitatem amat " and 
" Natura semper quod potest per faciliora non agit per ambages 
difficiles." Burtt has pointed out 1 that the decline of Aristotel- 
ianism in the sixteenth and seventeenth centuries coincided with 
a rise of neo-Platonism in which there was a strong Pythagorean 
element. For Kepler, in the extreme instance, the mathematical 
1 The Metaphysical Foundations of Modern Science (1932). 


harmony discoverable in the facts of Nature and even in 
the celestial regions was the reason why things are as they are. 
Nevertheless, this neo-Platonic mysticism was combined, in Kep- 
ler, with a reverence for exactitude in the mathematical 
formulation. Later this view was greatly modified; the 
existence of mathematical regularity was felt to indicate a 
mechanical explanation. The difficulty lay in combining the 
mathematical laws with a mechanism which would sacrifice none 
of their exactitude. Descartes's mechanism failed here. His cos- 
mology was in fact founded on an antithesis which was truly 
scholastic, since he tried to reconcile Copernicanism with an un- 
moved earth an earth at rest in its heaven. Huygens became 
convinced that Descartes in this and other matters had in fact 
repeated the errors of Scholasticism, for he hoped to found a de- 
monstrative and deductive system. Descartes, he noted, " who 
appears to me to have been very jealous of the renown of Galileo, 
had this great desire to pass for the author of a new philosophy. 
Which was clear from his efforts and his hopes to have it taught 
in the academies in place of that of Aristotle . . . ". Descartes's 
ideas, he admitted in another passage, were presented with all the 
force of verified conclusions; the novelty of the shapes of his 
ultimate particles of matter and the beauty of his vortices all 
exerted a compelling influence. " It seemed that when I read this 
book, the Principia, the first time that everything in the world 
became clearer, and I was sure that when I found some difficulty, 
that it was my fault that I did not understand his thought. I was 
then only fifteen or sixteen years old. But having since dis- 
covered from time to time things clearly false and others very 
improbable, I came back strongly to the preoccupation I was in, 
and at the present time I find scarcely anything I can accept as 
true in all the physics, metaphysics and meteors/' 

Huygens, in fact, returned to the outlook of Galileo. He had 
learnt that quantitative study of data and not scholastic logic fur- 
nished the technique of discovery. Descartes's powers as a mathe- 
matician compelled Huygens's admiration and he found it all the 
more unforgivable that exact agreement with the facts should 
not be uppermost in Descartes's work. The difference between 
Descartes and Huygens did not lie in their conception of the phy- 
sical processes so much as in the regard paid to accurate defini- 
tion of physical conditions. More than geometry was needed, 
Huygens perceived, to deal with this. There were more properties 


than mere extension, which was quite inadequate to account for 
the results of collision between elastic bodies and for accelerated 

These considerations are, however, as yet inadequate as an 
account of the revolution in Mechanics which we have been led 
to discuss. For the change in mental categories and the expres- 
sion of laws in corresponding mathematical form was combined 
with a profound change in the method of explanation involved 
in all scientific work and especially in Mechanics. In broad 
terms, the revolution against Aristotelianism was the rejection 
of Aristotle's final cause in favour of a scientific mode of his 
efficient cause; men asked not what purpose but what process lay 
behind phenomena. How radical this change was we can realize 
only if we remember that for Aristotle, with his biological out- 
look, all events were part of a natural process of fulfilment, of 
realization of what was innate. Things converged towards an 
appointed and necessary end; it was consequently more interest- 
ing to enquire what that end is, rather than what mechanism 
underlies events. In Dynamics all motion was motion to an 
appointed place; all nature bore the character of an innate im- 
pulse to movement. 

Movement and change set greater problems for the philoso- 
phy of Plato. Whereas movement was regarded as continuous by 
Aristotle, to Plato all change appeared as a succession of Forms 
and consequently movement must be essentially discontinuous. 
It may well be, therefore, that for an Aristotelian of the sixteenth 
century movement held less interest and less to be studied than 
it did for those who were influenced by the revival of neo-Platonic 

There was, however, a second reason for the revolt against 
Aristotle, whose authority so dominated the scholastics. Quite 
apart from the difference in the nature of cause itself, there was 
the objection that no knowledge of the world as it is could be 
derived by deductions from a priori metaphysical principles. 
Aristotle's system was out of touch with brute facts, his method 
unsuited to their investigation. Aristotle, in fact, did not see that 
induction demanded a correlating idea; that it could not 
be reduced to some kind of syllogism, and consequently that know- 
ledge of the world in the last resort cannot be made part of 
logic. Bacon, though not a scientist, was clear on this " unfruit- 
fulness" of Aristotle and expressed the new attitude to facts in his 


famous aphorism : " Nature to be commanded must be obeyed". 
It became clear from this time onwards that much remained to be 
discovered in respect of even the simplest events. 

Nevertheless, it is clear that what we now hold to be the 
typical method of the physical sciences appeared then to be a 
much more individual question. Galileo had given an excellent 
example but Huygens carried the inter-relation of mathematics 
and experiment a long way further. All his work illustrates this 
quality, and certain continental writers have even argued that 
Huygens's conception of scientific method was in some respects 
superior to that of Newton. Certainly in regard to the position 
of hypotheses in scientific work a case may be made out for 
Huygens's superiority. "... the main Business of natural Philo- 
sophy is to argue from Phaenomena without feigning Hypo- 
theses, and to deduce Causes from Effects, till we come 
to the very first Cause, which certainly is not mechanical," wrote 
Newton in the twenty-eighth query of his Optics. Against this 
we might place Huygens's remarks on the essential place of 
hypothesis in scientific work which come at the beginning of his 
Traite de la Lumiere. 

It was in the use of abstractions that Newton made the 
greatest contribution. He had the strongest objection to hypo- 
thetical entities because he wished to concentrate on mathematical 
relations. The space and time of Newton's system were not 
identical with the space and time of ordinary experience; they 
were abstractions. Huygens was unable to understand Newton's 
more positivist attitude here. The realist nature of his own con- 
ceptions makes him, in fact, a convenient starting point from 
which to trace the second main stream of scientific thought. 
Huygens might have accepted the term " correlate " as equivalent 
to hypothesis but, in the manner of all atomists, his concepts 
borrowed a garb from perceptual phenomena. For him the atom 
was a potential phenomenon, as it still is for some modern men 
of science. He did not see as clearly as Newton that for Science 
what is needed is a number of principles of quantitative correla- 
tion. This, surely, is the true end of scientific induction. In 
practice, in the scientific interpretation of phenomena not capable 
of treatment by classical mechanics, much use is always made of 
entities which must possess the qualities of hypotheses. This 
method Huygens, if not Newton, would have supported. We 
may summarize this discussion by saying that the Huygensian 


method confers greater freedom than the Newtonian and, in 
regard to the broad conception of the place of hypotheses, 
Huygens was a more profound methodologist than Newton. 

Huygens's work was very influential in its day and it is cer- 
tain that Newton was at least stimulated by him. An interest- 
ing example is Huygens's account of the ether setting aside, 
for the moment, the distinction of the subtle matter. Newton 
did not accept this sort of explanation of gravity but equally 
he recognized the difficulties of the so-called "action at a 
distance ". He believed with justification that his Principia dis- 
posed completely of the Cartesian vortices of subtle matter and 
he returned to this subject in his Optics. "... against fill- 
ing the heavens with fluid mediums, unless they be exceeding 
rare, a great objection arises from the regular and very lasting 
motions of the planets and comets in all manner of courses 
through the heavens. For thence it is manifest, that the heavens 
are void of all sensible resistance and by consequence of all sen- 
sible matter/' But Newton never denied that an ether might exist 
and that it might be conceived as a medium susceptible of trans- 
mitting vibrations. Burtt 1 even writes: " Halving taken over the 
notion from the current of the times, and feeling it to be thus 
well grounded, it was easy for Newton to extend its use to other 
phenomena which involved action at a distance and which others 
were accounting for in the same fashion, such as gravity, mag- 
netism, electric attraction, and the like." Newton was, however, 
not at all consistent on the subject of the ether and it is clear 
that his conception of it was different from that of Hooke. There 
is some probability that it was Huygens's work on the subject 
and particularly the experiments using the vacuum pump which 
influenced him most. Accounts of Huygens's experiments on 
the non-descent of columns of water and mercury were common 
in England, notably in the Philosophical Transactions, and the 
observations were also made by Boyle, Brouncker and others. 
Most of Newton's comments on the ether are to be found in his 
Optics, in which they first appeared in the Latin edition of 1706. 

The only acknowledged debt of Newton to Huygens was the 
statement of the theorems of centrifugal force in the Horologium 
Oscillatorium. Newton, as L. T. More has remarked, " must have 
seen that Huygens's law of centrifugal force was easily deduciblc 
from his own calculation on the attraction of the moon, and that 
* E. Burtt. op. cit. p. 165. 


by neglecting to follow up his work, Huygens had preceded him." 
This is suggested by Newton's statement : " What Mr. Huygens 
has published since about centrifugal forces I suppose he had 
before me." The view which is now most widely held in regard 
to the difficult problem of the order of Newton's ideas is that 
he solved the problem of centrifugal force independently of Huy- 
gens but neglected to publish anything until he had proved the 
important theorem concerning the gravitational field due to a 
large solid sphere. Newton was uninfluenced by Huygcns's very 
important development of the conceptions ot energy and work 
done in mechanical systems. An account of the growth of liuy- 
gens's ideas has been given in the sections on impact and on the 
centre of oscillation. 1 Unfortunately, the only direct influence 
of Huygens's ideas is to be found in the work of Leibriirzians, 
who made the conservation of vis viva a cosmic principle instead 
of treating it, as Huygens did, as only half the true law of the 
conservation of mechanical energy. Huygens did not go so far 
as Newton towards complete scientific positivism but he agreed 
that mathematical law is in itself the most important end of 
scientific work. For Newton the essential aim of science, in his 
own words, was to replace " occult properties supposed to result 
from specific Forms of Things " by " general laws of Nature ". 
The wisdom of this limited aim has been fully demonstrated. 

If Huygens saw the direction in which Leibnitz's ideas were to 
lead philosophy it is certain that he could not have sympathized 
with them. With his increasing recognition of Descartes's errors 
in purely physical matters, Huygens, as we know, paid less atten- 
tion to his philosophical system and so felt unperturbed by the 
dilemma of his dualism. If this had not been so, Huygens might 
have inclined to a materialism of the kind elaborated by his con- 
temporary, Hobbes, but it is inconceivable that he would have 
tolerated many of the latter's crudities. Huygens was something 
of a materialist and he rejected the orthodox religious doctrines: 
nevertheless, if he had written philosophy it would have been 
without the harshness of Hobbes's determinist schemes. 

Yet the bifurcation of the world into physical and mental 
spheres was as much the work of men like Huygens as it was of 
Descartes, who first presented the bifurcation in its most uncom- 
promising form. Kepler took the first step of distinguishing pri- 
mary from secondary qualities. For him only those qualities 

1 Nature, 1943, p. 519. 


which could be measured were primary and this attitude was 
found in Galileo, who defined the two classes more clearly and 
made the corresponding distinction between true knowledge (of 
primary qualities) and mere opinion. Colour, smell, taste and 
sound were for him matters of opinion subjective impressions 
resulting from the operations of atoms or vibrations on the sense 
organs. This set the stage for the Cartesian dualism. It is not 
perhaps properly recognized how far Huygens contributed to the 
adoption of Galileo's attitude. But we have only to recall that 
Huygens's whole work was the reduction of more phenomena to 
quantitative treatment their transformation from matters of 
opinion to those of knowledge. Scientific time measure for short 
intervals began with Huygens; he related the standard of length 
with that of time, he provided a geometrical treatment of reflec- 
tion and refraction of light, he greatly extended Mechanics and 
elucidated the true nature of more celestial phenomena in accord- 
ance with Copernicanism all this is a direct continuation of 
lines of thought to be found in Galileo. Through him, in fact, 
the main stream of scientific thought may be said to have been 
diverted from following Descartes and instead directed into the 
channel which Newton's work deepened to a river. But Huygens 
would not have had us forget the imaginative stimulus of Des- 
cartes 's writings. As late as 1691 he remarked: "We owe 
much to Descartes because he revealed new paths in the study 
of Physics and started the idea that everything must be reduced 
to mechanical laws/' 

Huygens came nearest to explaining his own method in 
scientific study in a letter to Tschirnhaus in 1687. ^ n thc problems 
of Physics, he wrote, very great difficulties are felt at first and 
these cannot be overcome " except by starting from experiments 
. . . and then by conceiving certain hypotheses . . . But even 
so very much hard work remains to be done and one needs not 
only great perspicacity but often a degree of good fortune." One 
is reminded of Whitehead's remark ". . . it is the establishment 
of the procedure of taking the consequences seriously which 
marks the real discovery of a theory ". At times Huygens 
followed his own procedure of taking the consequences seriously 
up to the point at which his elegant geometrical constructions 
were threatened. For him, as for Kepler, there seems to have 
been a disposition to believe that mathematical elegance is in 
some way an index to reality. This feeling for form may be 


found in many men of science but in few has it been so marked 
as in Huygens. This sense for elegant theoretical construction, 
roused by the symmetry and order which can be found in 
Nature, is well illustrated by the research on Iceland spar in 
which one has, as it were, the essence of Huygens's thought. Yet 
it was the weakness which predisposed him to believe in the 
rotation of the matiere subtile around the earth. He agreed with 
Leibnitz's comparison of Galileo and Descartes : " Galileo excels 
in the art of reducing mechanics to science; Descartes is admir- 
able at explaining by beautiful guesses the causes for the effects 
of nature." He agreed, too, with Leibnitz's remark that 
Descartes's work was " un beau roman de Physique " ; one feels 
that it was with regret. And yet, without Huygens's careful 
studies how could Leibnitz have coined that phrase? 


Many of the persons mentioned in the first part of this book will 
be unknown to the general reader, and some of them are but little 
known to the historian. The following notes provide some of the chief 
facts known about them. 

Auzout, Adrien (d. 1691). French mathematician and astronomer. 
Auzout is chiefly remembered for his invention of a micrometer con- 
sisting of movable hairs, mounted parallel in the field of view of the 
astronomical telescope. With Picard (below) he was a pioneer in the 
application of the telescope to graduated scales, thus replacing the use 
of open sights upon which Tycho Brahe had had to rely. He worked 
in Paris with Huygens and helped in the development of the 
" aerial " telescope. 

Barrow, Isaac (1630-77). This noted English mathematician was 
the first holder of the Lucasian professorship at Cambridge; on his 
arrival he became Newton's tutor and he undoubtedly influenced 
Newton through his own interests in geometrical optics and mathe- 
matics. Previously he had had a somewhat eventful life, his political 
and religious views as a young man in his twenties making it desirable 
that he should travel on the Continent. He became F.R.S. in 1662, 
the year before he went to Cambridge. In 1669 he resigned the 
Lucasian chair in Newton's favour, desiring among other things to 
devote himself to theology. 

Boulliau, Ismael (160594). French astronomer. Before working in 
Paris, where he met Huygens, he spent some time travelling in Italy, 
Holland and Poland, becoming acquainted with the leading men of 
science. He was a friend of Leopold de Medici and corresponded 
with him on scientific matters of the time. 

Campanella, Tomasso (1568-1639). He does not come into the 
period covered by this book, but his writings and opinions were well- 
known to Huygens and he is to be remembered as one of the earliest 
champions of the experimental method. His most famous work is his 
Defence of Galileo (1622), in which he quoted the great religious 
thinkers from Augustin to Aquinas to show how unjustified the 
persecution of Galileo was. 



Carcavy, Pierre de (d. 1684). Did important administrative work in 
Paris during the period under review. Colbert made Carcavy librarian 
to the King (1663) and it was during Carcavy's period of office that 
the royal library was moved to new rooms which became also the 
meeting place of the Academic Royale des Sciences. Carcavy's 
mathematical knowledge was good and he took part in 1645 * n tne 
dispute over the quadrature of the circle, claiming to demonstrate 
that the quadrature was impossible. He was a friend of Pascal, 
Descartes, Roberval and Fermat. 

Cassini, G. Domenico (1652-1712). Born at Perinaldo near Nice, 
and educated by the Jesuits at Genoa, Cassini at twenty-five became 
professor of astronomy at the famous university of Bologna. Here he 
became distinguished and his fame spread to Paris when he published 
ephemerides for the satellites of Jupiter. Through the efforts of Picard 
he was invited to Paris and after 1671 he was virtually the director of 
the Paris observatory. He collaborated with Richer in 1672 on the 
observations of the parallax of Mars. With Picard and Auzout he 
did a great deal towards founding the great tradition of the Paris 
observatory. His most famous observation was the discovery of the 
satellites of Saturn and the division in its ring now known by his name. 

Fermat, Pierre de (1601-65). One of the greatest of French mathe- 
maticians. Fermat anticipated Descartes's invention of analytical 
geometry in nearly all respects, and he contributed important work on 
the treatment of maxima and minima. With Pascal he laid the 
foundations of the mathematical study of probability (Huygens also 
wrote on this subject), and he discovered the important " least-time 
principle " in optics. He adversely criticized some of Descartes's 
optical theory. 

Frenicle de Bessy (1605? 1675). Frenicle held an official position in 
Paris, where he acquired a great reputation as a demon calculator. He 
could solve the most complex numerical problems in arithmetic and 
so quickly as to astonish even Fermat, Roberval and Wallis. His 
method of working was kept a secret and was examined after his 
death. His " method of exclusions ", which he used so much, is now 
only an object of curiosity and he contributed nothing of permanent 

Hevelius(or,]ohann Hevel) (1611-87). Noted astronomer of Danzig. 
Although he made first-class observations of the planets and the moon 
employing telescopes, he preferred to make all his measurements 
with open sights. This led to a dispute with the English astronomers 
and he was visited in Danzig by Halley, who observed with him and 
employed telescopic sights. Neither astronomer converted the other. 


Mariotte, Edm& (1620-84). A Roman Catholic abbe with a weak- 
ness for experimentation. He joined the Academic Royale des 
Sciences in the year of its foundation. His most valuable work was 
on impact; he independently enunciated the law known as Boyle's 
law fcf. his Discours de la Nature de VAir (1676)]. 

Papin, Denis (16471712). Had a rather precarious existence and 
died in obscurity in London. For a time he assisted Huygens, notably 
with his experiments on the air pump and on the expansion of steam 
as a source of power. Later he came to London to work with Boyle 
and in 1684 he became temporary curator to the Royal Society. In 
1687 he was appointed professor of mathematics at Marburg. From 
there he went to Cassel (1696) and then on to London (1707). 

Petit, Pierre (1594-1677). An able civil servant who was given his 
chance by Richelieu. He was quite a distinguished mathematician 
and he had quite a lot of contact with Huygens during the latter's 
early visits to Paris. His daughter was attractive. A friend of Pascal 
also, he helped him in his experiments on the vacuum. 

Picard, Jean (1620-82). One of the most famous astronomers ot 
the seventeenth century. He first of all observed with Gassendi, whom 
he replaced as professor of astronomy at the College de France. In 
1666 he became an abbe* but without any interruption of his scientific 
output. In 1669 he read to the Academic an important memoir on the 
new methods in astronomy resulting from work by Huygens and 
himself. He went to Uraniborg, Tycho Brahe's famous observatory 
(of which nothing then remained), in order to fix more accurately its 
latitude and longitude. He met Roemer on this journey and got him 
to come back to Paris with him. Picard's Mesure de la Terre was an 
important work and provided Newton with the information he needed 
in his calculations on the earth's gravity. 

Roberval, Giles Persone de (1602-75). A French mathematician 
who rose to eminence from obscure origins. He came to Paris in 1627 
and stayed with Mersenne. In 1631 he was appointed to a chair of 
philosophy and then to the professorship of mathematics at the 
College de France. He was one of the original members of the 
Academic Royale des Sciences. After 1638 he had strong differences 
with Descartes and became hostile to Descartes's philosophy. He was 
a man of decided originality of mind and his mathematical work is 
of importance. 


Roemer, Glaus (16441710). Noted Danish astronomer. Roemer 
worked at first at the Round Tower of Copenhagen, which had been 
built by Christian IV for one of Tycho Brahe's assistants. He is 
chiefly remembered for bringing the transit telescope into general 
use. His calculation of the velocity of light is also famous. After his 
death his instruments and records were destroyed in the great fire of 
Copenhagen (1728), but it was found possible to reconstruct much 
of what was lost. 

Sorbiere, Samuel de (1615-70). French writer. Sorbiere was in- 
tended at first for the Protestant ministry and was brought up by an 
uncle after the death of his parents. He studied medicine in Paris 
and practised in Holland up to about 1650. In 1653 ^ e became a 
Catholic, and in 1655 he went to Rome and was received by the Pope. 
Lacking any more solid rewards, Sorbiere sought patronage in Paris 
and became loosely attached to the circle of scientific amateurs. He 
was not thought to have much originality or learning by his 

Wallis,John (1616-1703). Famous English mathematician. Wallis 
was ordained in the English Church after completing his time at 
Emmanuel College, Cambridge, and he then served as chaplain to a 
noble family. His fortune was made by his discovery of his ability to 
decipher codes for the parliamentary party during the civil war. 
In 1649 Cromwell appointed him Savilian professor of Geometry at 
Oxford. He became a friend of Newton and his Arithmetica 
Inftnitorum led the latter direct to his discovery of the binomial 
theorem. He became associated with Boyle and helped to found the 
Royal Society. 



There is a wealth of information in Huygens's correspondence 
contained in Volumes i to 10 inclusive of the Oeuvres Completes de 
Christiaan Huygens, published by the Socie'te Hollandaise des 
Sciences. The notes given in Part I of this book represent a very small 
selection of this mass of information. 

Also very useful for the student is Le Sejour de Christian 
Huygens a Paris by H. L. Brugmans (1935). 



A History of Science, Technology and Philosophy (i6th and I7th 
Centuries) by A. Wolf. (Allen and Unwin, 1935). 

The Metaphysical Foundations of Modern Science by E. Burtt. 
(Kegan Paul, 1932). 

Ceschichte der Physik by J. C. Poggendorff, (1879). 

The Role of Scientific Societies in the Seventeenth Century by 
M. Ornstein (Chicago, 1928). 

Scientific Organisations in Seventeenth Century France by H. 
Brown (Baltimore, 1934). 


Le Developpement de la Physique Carttsienne, 1646-1712 by P. 
Mouy (Paris, 1934). 

Isaac Newton by L. T. More (Scribners, 1934). 

Les Origines de la Statique by P. Duhem (1905). 

La Theorie Physique by P. Duhem (1906). 

The Science of Mechanics by E. Mach (English trans, by T. J. 
McCormack, 1919). 

The Mathematical Principles of Natural Philosophy by I. 
Newton (English trans, by F. Cajori, 1934). 

Discourses on Two Sciences by G. Galilei (English trans, by 
Crew and De Salvio, 1914). 

A History of the Theories of /Ether and Electricity by E. T. 
Whittaker (1910). 

Optics by L Newton (Bell, 1931). 

Matter and Gravity in Newton's Physical Philosophy by A. J. 
Snow (1926). 



Die. Pendeluhr by Christian Huygens (German trans, by 
Heckscher and Oettingen, 1913), Ostwald's Klassiker der Exakten 
Wissenschaften No. 192. 

Oeuvres Completes, vols. 16, 17, 18, 19. 

The Evolution of Clockwork by J. Drummond Robertson (1931). 


Treatise on Light by Christian Huygens (English trans, by S. 
Thompson, 1912). 

Oeuvres Completes, vols. 13, 19. 

The Principles of Physical Optics by E. Mach (English trans, by 
Anderson and Young, 1926). 


Oeuvres Completes, vols. 15, 21. 

Histoire de lAstronomie Moderne by A. Delambre (1821). 

This bibliography is limited to comparatively modern publica- 
tions. The references to the Oeuvres Completes de Christian H 
Huygens are not intended to be complete, but they indicate the 
volumes in which the greater part of a particular subject is treated. 
There is, in addition, the work on Mathematics to he found in vols. 
u, 12. 14, 20 



54, 58, 78 

Accademie del Cimento, 36, 203 
Anagrams for publication of discov 

cries, 32 

Aristotelianism, 52, 97 
Atomic theory, 28, 94, 101 
Auzout. Adrien, 41, 57, 59, 213 


Barrow, Isaac, 65, 213 

Benedetti, 21, 99 

Boulliau, Ismael, 27, 32, 210, 

Boyle, Robert, 48, 57, 9:? 

Brahe, Tycbo, 35, 127 

Breda, College of, 20 

Bruce, Alexander, Earl of Kincardine, 

Campani, Guiscppe, 44, 56 
Cartesianism, 15, 20, 24, 60, 68, 8, 

86, 109 
Cassini, Domenico, 40, 43, 56, 63, 77 

102, 214 

Centre of gravity, 112, i*5> 14^ 
Centre of oscillation, 150 
Centre of percussion, 22, 150 
Centrifugal force, 64, 117, 161 
Chapelain, Jean, 27 
Chemistry, 93 
Chromatic aberration, 173 
Circular motion, 63, 85 
Clock, pendulum, 34, 127; in astron 

omy, 39, 133; in determining longi 

tudes. 40, 54, 83, 89, 135; spring 

regulated, 55, 70 
Colbert, 6, 41/46, 83 
Colour, 57, 71 
Comets, 89 

Compound pendulum, 99, 150 
Conical pendulum, 69, 120 
Conrart, 27 

Conservation of energy, 114, 154 
Conservation of momentum, no 
Copernicanism, 34, 44, 99, 193 
Cosmotheoros, 89, 200 
Coster, Samuel, 37 
Cvcloidal pendulum, 37, 39, 136 

De Motu ex Pefcusu'one, i 1 1 

De Vi Ccntrifuga, 117 

Descartes, Ren6, 14, 18, ioj, 110. 176 

Diffraction, 174 

Divinis, Eustachio de, 34, 44 

Double refraction, 74, 183 


Eclipses, of sun, 62 

Energy, conservation of, 114, 154 

Ether, 162, 177, 180, 208 

Evolutes of curves, 146 

Eyepiece, Huygens's, 52, 171 

Fermat. Pierre de, 81, 182. 214 
Fermat's principle, 58, 182 

GALILEO, 22, 30, 98, 136, 150 
Gassendi, Pierre, 27, 105 
Gilbert, William, 101, 161 
Gravity, 33, 63, 85, 90, 117, 161 
Gresham College, 41, 44, 54 

HEVELIUS, JOHANN, 30, 62, 214 

Hobbes, Thomas, 46, 209 

Holmes, Captain, 51 

Hooke, Robert, 55,' 70, 73, 144; Micto- 
graphia, 57, 65, 176; wave theory, 
77, 176 

Horo/ogmm, 35, 128 

Horologium Oscillatorium, 40, 61, 67, 
131, 138 

Huygens, Christian, character, 52, 60; 
education, 7, 19; health, 7, 60, 65, 
75 94 as mathematician, 25, 68, 
81, 91; conception of scientific- 
method, 73, 178, 207, 210; mechan- 
istic outlook, 9, 79, 162; and New- 
ton, 15, 71, 72, 84, 209; and Leib- 
nitz, 86, 88; and Hooke, 55, 70, 73: 
and Spinoza, 9, 62; and Bayle, 87; 
and Huet, 87; in Paris, 7, 29, 43, 
58, 67, 77; in London, 45, 50, 53, 
84; religious outlook, 7, 94, 202; on 
universal gravity, 86, 89; atomic 
theory, 28, 94; and reflecting tele- 
scope, 71, 93; eyepiece, 52, 171; 
work in optics, 74, 81, 167 

Huygens, Constantin (the elder), 16: 
(the younger). 19, 9* 




ICELAND SPAR, 74, 177, 184 

Impact, theory of, 64, 109 
Intercepted pendulum experiment, 

Journal des Savants, 55 
Jupiter, 36 

KEPLER, 98, 165 

Leibnitz 69, 80, 86, 88, 90 
L ens-grindiug, 51 
Leopold de Medicis, 33, 35 
Lcyden University, 19 
Light, propagation of, 76, 176 
Longitude determination, 35, 89, 13* 
Louis XIV, 47, 61 

MARIOTTE, EDM, 64, 164, 215 

Mars, 103 

Mersennc, Marin, 21, 25, 100, 138 

Micrometer, 57, 197 

Microscope, 65, 82, 174 

Moment of inertia, 151 

Momentum, 64 

Montmor Society, 41, 43, 53 

NEWTON, and centrifugal force, 64, 
09; the Principta, 84, 89; and the 
ether, 85, 208; and the Horologium 
Oscillatorium, 68; and Huygens, 15, 
71, 72, 84; and his reflecting teles 
cope, 70; and colours, 71 

Oldenburg, Henry, 45, 53, 65, 71, 78 

PAPIN, DFNLS, 67, 74, 190, 215 
Pascal, 23, 38, 43, 147 
Philosophical Transactions, w 
Picard, Jean, 59, 67, 103, 215 

Planetary machine, 83 
Polarized light, 189 
Potver, Thomas, 8 

RtFRAMIVE INDEX, 167, l8l 

Richer, 103 

Roberval, Giles Persone de, 25, 32, 

164, 215 

Roemer, Olaus, 39, 77, 83, 216 
Renault, Jacques, 95, 116 
Royal Society, 41, 53, 66 

SATURN, 26, 30, 44, 52, 193 

Schootcn, Frans van, 19, 25 

Simple hiirmonic motion, 99, 143 

Sorbierc, Samuel dc, 28, 41, 53, 216 

Space, absolute, 122 

Spherical aberration, 171 

Spino/a, 9, 62 

Statics, 124 

Stevin, Simon, 19, 21 

Subtle matter, 49, 163, 178 

Sy sterna Saturniurn, 33, 195 


Telescopes, 26, 30, 45, 56, 193; aerial, 

51, 56 

ThcVenot, Melchisc'dech, 41, 54 
Thuiet, 38 

TraitS dc la Lumiere, 49, 76, 81, 178 
Tschirnhaus, Walter von, 75 

VACUUM, 23, 163 

Viviani, 23 

Voorbutg, 55 

Vortex theory, 24, 86, 90, 109, 163, 201 

WALLIS, JOHN, 45, 115, 126, 216 
Wave theory of light, 76, 77, 176 
Weight and mass, 119 
Wren, Christopher, 52